Post on 25-Feb-2016
description
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 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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
(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
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