Identity amp Inverse Matrices

Section 4-7

Identity Matrices

In the multiplication of numbers the identity element is the number 1

since x 1 = x for every value of x

(it gives the original number its identity back)

If you multiply the matrix I with any matrix P and the result is the matrix P then I is known as the identity matrix

For matrices the number 1 is 1 0 01 0

0 1 00 1

0 0 1or

Multiplicative Identity

Is there a 2 2 identity matrix for matrix multiplication ie

1 00 1

A I = I A = A where I is the identity matrix

For example I =

dbca

dbca

()

NB When referring to the multiplicative identity it is usually called the identity matrix

Is is a square matrix All elements in the leading diagonal

are 1 All the other elements are 0 Eg

Characteristics of Identity matrix

1 0 01 0

0 1 0 0 1

0 0 1etc

What do you obtain when A is multiplied by the identity matrix

1 00 1

a b a bc d c d

AI = A or IA = A

When we say the inverse of a matrix it is referring to the multiplicative inverse

Matrix Inverse

2 3 3 3If A = and B =

1 4 5 2

2 3 3 3 1 0then AB = I and BA= I

1 4 5 2 0 1

If A and B are two matrices and AB = BA = I then A is said to be the inverse of B denoted by B-1B is said to be the inverse of A denoted by A-1

Given A and the inverse of A denoted by A-1

IMPT NOTE if two matrices are inverses and you multiply them then the result is the IDENTITY MATRIX

-1

-1

AA I

A A I

Step 1 Find the determinant of the matrix A denoted by det A

det A = a b

ad bcc d

To find the inverse of a matrix A = a bc d

Step 2 The inverse of matrix A is

Note bull If det A = 0 then the inverse of A is not defined bull Hence A does not have an inverse

1det

d bc aA

Find the inverse if it exists

Find the inverse if it exists

Examples3 14 1

1 114 33 1 1( 4)

1 114 37

1 17 7

347 7

6 38 4

6 318 46 4 3 4

6 318 40

Impossible

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Identity Matrices

In the multiplication of numbers the identity element is the number 1

since x 1 = x for every value of x

(it gives the original number its identity back)

If you multiply the matrix I with any matrix P and the result is the matrix P then I is known as the identity matrix

For matrices the number 1 is 1 0 01 0

0 1 00 1

0 0 1or

Multiplicative Identity

Is there a 2 2 identity matrix for matrix multiplication ie

1 00 1

A I = I A = A where I is the identity matrix

For example I =

dbca

dbca

()

NB When referring to the multiplicative identity it is usually called the identity matrix

Is is a square matrix All elements in the leading diagonal

are 1 All the other elements are 0 Eg

Characteristics of Identity matrix

1 0 01 0

0 1 0 0 1

0 0 1etc

What do you obtain when A is multiplied by the identity matrix

1 00 1

a b a bc d c d

AI = A or IA = A

When we say the inverse of a matrix it is referring to the multiplicative inverse

Matrix Inverse

2 3 3 3If A = and B =

1 4 5 2

2 3 3 3 1 0then AB = I and BA= I

1 4 5 2 0 1

If A and B are two matrices and AB = BA = I then A is said to be the inverse of B denoted by B-1B is said to be the inverse of A denoted by A-1

Given A and the inverse of A denoted by A-1

IMPT NOTE if two matrices are inverses and you multiply them then the result is the IDENTITY MATRIX

-1

-1

AA I

A A I

Step 1 Find the determinant of the matrix A denoted by det A

det A = a b

ad bcc d

To find the inverse of a matrix A = a bc d

Step 2 The inverse of matrix A is

Note bull If det A = 0 then the inverse of A is not defined bull Hence A does not have an inverse

1det

d bc aA

Find the inverse if it exists

Find the inverse if it exists

Examples3 14 1

1 114 33 1 1( 4)

1 114 37

1 17 7

347 7

6 38 4

6 318 46 4 3 4

6 318 40

Impossible

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Multiplicative Identity

Is there a 2 2 identity matrix for matrix multiplication ie

1 00 1

A I = I A = A where I is the identity matrix

For example I =

dbca

dbca

()

NB When referring to the multiplicative identity it is usually called the identity matrix

Is is a square matrix All elements in the leading diagonal

are 1 All the other elements are 0 Eg

Characteristics of Identity matrix

1 0 01 0

0 1 0 0 1

0 0 1etc

What do you obtain when A is multiplied by the identity matrix

1 00 1

a b a bc d c d

AI = A or IA = A

When we say the inverse of a matrix it is referring to the multiplicative inverse

Matrix Inverse

2 3 3 3If A = and B =

1 4 5 2

2 3 3 3 1 0then AB = I and BA= I

1 4 5 2 0 1

If A and B are two matrices and AB = BA = I then A is said to be the inverse of B denoted by B-1B is said to be the inverse of A denoted by A-1

Given A and the inverse of A denoted by A-1

IMPT NOTE if two matrices are inverses and you multiply them then the result is the IDENTITY MATRIX

-1

-1

AA I

A A I

Step 1 Find the determinant of the matrix A denoted by det A

det A = a b

ad bcc d

To find the inverse of a matrix A = a bc d

Step 2 The inverse of matrix A is

Note bull If det A = 0 then the inverse of A is not defined bull Hence A does not have an inverse

1det

d bc aA

Find the inverse if it exists

Find the inverse if it exists

Examples3 14 1

1 114 33 1 1( 4)

1 114 37

1 17 7

347 7

6 38 4

6 318 46 4 3 4

6 318 40

Impossible

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Is is a square matrix All elements in the leading diagonal

are 1 All the other elements are 0 Eg

Characteristics of Identity matrix

1 0 01 0

0 1 0 0 1

0 0 1etc

What do you obtain when A is multiplied by the identity matrix

1 00 1

a b a bc d c d

AI = A or IA = A

When we say the inverse of a matrix it is referring to the multiplicative inverse

Matrix Inverse

2 3 3 3If A = and B =

1 4 5 2

2 3 3 3 1 0then AB = I and BA= I

1 4 5 2 0 1

If A and B are two matrices and AB = BA = I then A is said to be the inverse of B denoted by B-1B is said to be the inverse of A denoted by A-1

Given A and the inverse of A denoted by A-1

IMPT NOTE if two matrices are inverses and you multiply them then the result is the IDENTITY MATRIX

-1

-1

AA I

A A I

Step 1 Find the determinant of the matrix A denoted by det A

det A = a b

ad bcc d

To find the inverse of a matrix A = a bc d

Step 2 The inverse of matrix A is

Note bull If det A = 0 then the inverse of A is not defined bull Hence A does not have an inverse

1det

d bc aA

Find the inverse if it exists

Find the inverse if it exists

Examples3 14 1

1 114 33 1 1( 4)

1 114 37

1 17 7

347 7

6 38 4

6 318 46 4 3 4

6 318 40

Impossible

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

When we say the inverse of a matrix it is referring to the multiplicative inverse

Matrix Inverse

2 3 3 3If A = and B =

1 4 5 2

2 3 3 3 1 0then AB = I and BA= I

1 4 5 2 0 1

If A and B are two matrices and AB = BA = I then A is said to be the inverse of B denoted by B-1B is said to be the inverse of A denoted by A-1

Given A and the inverse of A denoted by A-1

IMPT NOTE if two matrices are inverses and you multiply them then the result is the IDENTITY MATRIX

-1

-1

AA I

A A I

Step 1 Find the determinant of the matrix A denoted by det A

det A = a b

ad bcc d

To find the inverse of a matrix A = a bc d

Step 2 The inverse of matrix A is

Note bull If det A = 0 then the inverse of A is not defined bull Hence A does not have an inverse

1det

d bc aA

Find the inverse if it exists

Find the inverse if it exists

Examples3 14 1

1 114 33 1 1( 4)

1 114 37

1 17 7

347 7

6 38 4

6 318 46 4 3 4

6 318 40

Impossible

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Given A and the inverse of A denoted by A-1

IMPT NOTE if two matrices are inverses and you multiply them then the result is the IDENTITY MATRIX

-1

-1

AA I

A A I

Step 1 Find the determinant of the matrix A denoted by det A

det A = a b

ad bcc d

To find the inverse of a matrix A = a bc d

Step 2 The inverse of matrix A is

Note bull If det A = 0 then the inverse of A is not defined bull Hence A does not have an inverse

1det

d bc aA

Find the inverse if it exists

Find the inverse if it exists

Examples3 14 1

1 114 33 1 1( 4)

1 114 37

1 17 7

347 7

6 38 4

6 318 46 4 3 4

6 318 40

Impossible

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Step 1 Find the determinant of the matrix A denoted by det A

det A = a b

ad bcc d

To find the inverse of a matrix A = a bc d

Step 2 The inverse of matrix A is

Note bull If det A = 0 then the inverse of A is not defined bull Hence A does not have an inverse

1det

d bc aA

Find the inverse if it exists

Find the inverse if it exists

Examples3 14 1

1 114 33 1 1( 4)

1 114 37

1 17 7

347 7

6 38 4

6 318 46 4 3 4

6 318 40

Impossible

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Find the inverse if it exists

Find the inverse if it exists

Examples3 14 1

1 114 33 1 1( 4)

1 114 37

1 17 7

347 7

6 38 4

6 318 46 4 3 4

6 318 40

Impossible

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Determine whether each pair of matrices are inverses

Examples

32

2 12 213 4

and

If 2 matrices are inverses when you multiply them you get the identity matrix

1 00 1

32

2 12 213 4

1 00 1

Yes ndash

theyrsquore

inverses

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

To solve simultaneous equations by using simple algebra if there is no solution or infinite solutions what will you say about the two equations

The simultaneous equations will represent either two parallel lines or the same straight line

Using Matrices to Solve Simultaneous Equations

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

When the simultaneous equations is expressed in the matrix form and if the determinant of the 22 matrix is zero then the two simultaneous equations will represent either two parallel lines or the same straight line

The equations have no unique solution

Using Matrices to Solve Simultaneous Equations

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Using Matrices to Solve Simultaneous Equations

bull Step 1 Given ax + by = h

and cx + dy = k a b x hc d y k

bull Step 2 Find determinant of

a bc d

bull Step 3 If

0a bc d

then 1x d b hy c a kad bc

bull Step 3 If

0a bc d

the equations have no unique solution

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Class work Q1 3 58 Q10 Q12 Q13 Q14

Ex 9D Page 214

bull Homeworkbull Q2 4 6bull Q9bull Q11

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Why learn Matrices The interior design company is given the job of putting up the curtains for the windows sliding doors and the living room of the entire new apartment block of the NTUC executive condominium There are a total of 156 three-bedroom units and each unit has 5 windows 3 sliding doors and 2 living rooms Each window requires 6 m of fabric each sliding door requires 14 m of fabric and each living room requires 22 m of fabric Given that each metre of the fabric for the window cost $1230 the fabric for the sliding door costs $1450 per metre and each metre of the fabric for the living room is $1650We can write down three matrices whose product shows the total amount of needed to put up the curtains for each unit of the executive condominium

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

NE Message

The property market in Singapore went up very rapidly in the 1990rsquos Many Singaporeans dream of owning a private property were dashed and many call for some form of help from the government to realise their dream NTUC Choice Home was set up to go into property business as a way of stabilising the market and to help Singaporeans achieve their dream of owning private properties With the onset of the Asian economic crises the property market went under and the public start to question the need for NTUC Choice Home and urged NTUC to dissolve NTUC Choice Homes Do you think this is a good request How long do you think it will take to set up a company to run the property business

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

The Microsoft Excel matrix functions are MDETERM(array) Returns the matrix determinant of an array MINVERSE(array) Returns the inverse of the matrix of an array MMULT(array A array B) Returns the matrix product TRANSPOSE(array) Returns the transpose of an array The first row of the

input becomes the first column of

the output array etc Except for MDETERM() these are array functions

and must be completed with Crtl+shift+Enter

Operations using a Spreadsheet

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Routes matrices or Matrices for Graphs Matrices can be used to store data about graphs

The graph here is a geometric figure consisting of points (vertices) and edges connecting some of these points If the edges are assigned a direction the graph is called directed

Cryptography Matrices are also used in cryptography the art

of writing or deciphering secret codes

Some Interesting Applications

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Routes Matrices

A E

D

C

BTo

A B C D EA 0 1 2 0 1

From B 1 2 0 0 1C 1 0 0 1 2D 1 0 1 1 1E 0 1 1 1 0

the loop at B gives 2 routes from B to B but the loop at D givesonly 1 route because it is one-way only

0111011101210011002110210

R =

Example If 5 places A B C D E are connected by a road system shown in the graph The arrows denote one-way roads then this can be listed as

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Multiplying this matrix by itself gives R2 which gives the number of possible two-stage routes from place to place Eg the number in the 1st row 1st column is 3 showing there are 3 two-stage routes from A back to A (One is ABA another is ACA using the two-way road and the third is ACA out along the one-way road and back along the two-way road)

Similarly R3 gives the number of possible three-stage routes from place to place and vice versa

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

A spreadsheet can be used for the tedious matrix operations as shown below

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

One way of encoding is associating numbers with the letters of the alphabet as show below This association is a one-to-one correspondence so that no possible ambiguities can arise

Cryptography

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

In this code the word PEACE looks like 11 22 26 24 22

Suppose we want to encode the message MATHS IS FUN

If we decide to divide the message into pairs of letters the message becomes MA TH IS SF UN

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

(If there is a letter left over we arbitrarily assign Z to the last position) Using the correspondence of letters to numbers given above and writing each pair of letters as a column vector we obtain

Choose an arbitrary 2 2 matrix A which has an inverse A-1 Say A = and

A-1 =

2614

AM

197

HT

188

IS

218

FS U 6

N 13

2132

2132

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23

Now transform the column vectors by multiplying each of them on the left by A

The encoded message is 106 66 71 45 70 44 79 50 51 32

To decode multiple by A-1 and reassigning letters to the numbers

• Identity amp Inverse Matrices
• Identity Matrices
• Slide 3
• Characteristics of Identity matrix
• Matrix Inverse
• Slide 6
• Slide 7
• Examples
• Examples (2)
• Using Matrices to Solve Simultaneous Equations
• Using Matrices to Solve Simultaneous Equations (2)
• Slide 12
• Ex 9D Page 214
• Slide 14
• Slide 15
• Operations using a Spreadsheet
• Some Interesting Applications
• Routes Matrices
• Slide 19
• Slide 20
• Cryptography
• Slide 22
• Slide 23