Basic Operations MultiplicationDeterminants Cramer’s RuleIdentityInverses Solving Systems of...

Basic Operations

Multiplication Determinants

Cramer’s Rule Identity Inverses

Solving Systems of equations

APPENDIX

MATRIX:

A rectangular array of numbers.

15 8 11 4

2 0 9 2

3 1 3 5

What is a matrix?

MATRICES: BASIC OPERATIONSMenuAppendix

DONUTS EATEN

Mon Tues Wed Thurs

Tom 15 8 11 4

Jerry 2 0 9 2

Steve 3 1 3 5

This matrix has… 3 rows and 4 columns.

43

We describe a matrix by It’s ROWs and COLLUMNs

We need to know the dimensions of a matrix before we can do much with it.

What is a matrix?A rectangular array of numbers. A axb matrix has “a” rows and “b” columns.

3x4

15 8 11 4

M 2 0 9 2

3 1 3 5


34333231

24232221

14131211

MMMM

MMMM

MMMM

3x4M

First column

First row

Second row

Third row

Second column

Third column

Fourth column

12M Row numberColumn number



15 8 11 4

2 0 9 2

3 1 3 5

What is entry 23?What is entry 32?What is entry 43?What is entry 34?

1 2 4 1 3 10

A 3 0 7 B 3 1 0

9 1 5 1 0 6

If A and B are two matrices of the same size (same dimensions),


C A B

0 5 14

6 1 7

10 1 11

then the sum of the matrices is a matrix C=A+B whose

entries are the sums of the corresponding entries of A and B

• Addition, Subtraction

hdgc

fbea

hg

fe

dc

ba

hdgc

fbea

hg

fe

dc

ba

dhcfdgce

bhafbgae

hg

fe

dc

ba

Just add elements

Just subtract elements

Multiply each row by each column


Properties of matrix addition:

1. Matrix addition is commutative (order of addition does not matter)

2. Matrix addition is associative

3. Addition of the zero matrix

ABBA

CBACBA

AA00A



7 3 1 5

5 8 4 1

5 2 6 5

9 7 2 3

3 1 1 9

5 28 4 7

9 72 1 1

3 1

Add the following:

8 8

1 9

1 3

7 4

2 8

UNDEFI NED

Multiplication by a scalar

If A is a matrix and c is a scalar, then the product cA is a matrix whose entries are obtained by multiplying each of the entries of A by c

1 2 4

A 3 0 7 c 3

9 1 5

MATRICES: MULTIPLICATIONMenuAppendix

cA

3 6 12

9 0 21

27 3 15

6 3 5 11

8 2 8 10

y x

z z

9

6

4

x

y

z

3 5 8 4 142

1 6 1 2

x

z y

14

1

2

x

y

z


Matrix multiplication is associative

Distributive law

Multiplication by identity matrix

Multiplication by zero matrix

CABBCA

AIA A;AI

00A 0;A0

CABAACB

ACABCBA


4

1

2

2

3

7

3

2

8

6

4

3

1

0

2

9

7

5

Here’s how multiplying matrices works:

5

2

24

78

57

35

52

44

45

24118

16014

12210

12363

8049

6635

18272

12056

9440


1

2

6

9

7

2

9

3

2

6

2

4

THE RULES:

The number of columns in the

first matrix must be the same as the number of

rows in the second

2 x 3 3 x 2

25

37

99

77 The resulting matrix has the

dimensions of the outer numbers

2 x 2


1

2

6

9

7

2

2

7

3

4

A) Can you multiply these matrices?B) If so, what are the dimensions of the resulting matrix?

1

2

6

9

7

2

1

0

2

9

5

4

3

8

1

5

3

5

2

7

3

4

1

2

6

9

7

2

4

2

7

6

3

4

3

2

7

9

5

1

4

2

7

6

3

4

0

1

6

2

7

9

8

4

9

4

5

1

2 x 2 3 x 2 2 x 2 2 x 3 2 x 3 3 x 2

3 x 2 3 x 2 3 x 2 2 x 32 x 2 4 x 2

2 X 32 X 2

3 X 3


Weird Matrix Thing:

1

3

2

6

5

1

2

4

5

1

2

4

1

3

2

6

If A and B are Matrices, A x B is not the same as B x A

11

13

2

22

3

21

6

30


MORE MATRIX RULES

ASSOCIATIVE PROPERTY A(BC) = (AB)C

If A B and C are Matrices…

ASSOCIATIVE PROPERTY (scalar) c(AB) = (cA)B = A(cB)DISTRIBUTIVE PROPERTY (kinda)

Left Distributive Prop. A(B+C) = AB + AC

Right Distributive Prop. (A+B)C = AC + BC


5 3 7 1

2 2 0 8

Practice Problems

3 17 1

2 40 8

0 2


35 29

14 14

21 11

14 34

0 16

2 x 2 2 x 2

2 X 2

3 x 2 2 x 2

3 X 2

4 3 9 0

5 3 7 1

2 2 0 8

Practice Problems

2 41 2 5

2 11 0 2

0 5


UNDEFI NED

6 31

2 6

The determinant of a square matrix is a numberobtained in a specific manner from the matrix.

For a 1x1 matrix:

For a 2x2 matrix:

What is a determinant?

1111 aA aA )det(;

211222112221

1211 aaaaA aa

aaA

)det(;

Product along red arrow minus product along blue arrow

MATRICES: DETERMINANTSMenuAppendix

211222112221

1211 aaaaA aa

aaA

)det(;

Product along red arrow minus product along blue arrow


6 31

2 6

Find the determinant

36 62

3662

98

The determinant of a square matrix is a numberobtained in a specific manner from the matrix.

What is a determinant?


What if the matrix isn’t square??

Then there is NO DETERMINANT.

6.837 Linear Algebra Review

Determinant of a Matrix• Used for inversion• If det(A) = 0, then A has

no inverse• Can be found using

factorials, pivots, and cofactors!

• Lots of interpretations – for more info, take 18.06

dc

baA

bcadA )det(

ac

bd

bcadA

11


75

31A

8537175

31)A det(

Consider the matrix

Notice (1) A matrix is an array of numbers(2) A matrix is enclosed by square brackets

•The determinant of a matrix is a number•The symbol for the determinant of a matrix is a pair of parallel lines

Computation of larger matrices is more difficult


For ONLY a 3x3 matrix write down the first two columns after the third column

11 12 13

21 22 23

31 32 33

a a a

| | a a a

a a a

A

Sum of products along red arrow minus sum of products along blue arrow

This technique works only for 3x3 matrices332112322311312213

322113312312332211

aaaaaaaaa

aaaaaaaaa)A

det(

aaa

aaa

aaa

A

333231

232221

131211


11 12

21 22

31 32

a a

a a

a a

2

01A

21-

4

3-42 2 4 3

1 0 4

2 1 2

0 32 30 -8 8

Sum of red terms = 0 + 32 + 3 = 35

Sum of blue terms = 0 – 8 + 8 = 0

Determinant of matrix A= det(A) = 35 – 0 = 35


Find the determinant

2 4

1 0

2 1

MATRICES: CRAMER’s RULEMenuAppendix

MATRICES: IDENTITYMenuAppendix

Zero matrix: A matrix all of whose entries are zero

Identity matrix: A square matrix which has ‘1’ s on the diagonal and zeros everywhere else.

0000

0000

0000

430 x

100

010

001

33xI

MATRICES: IDENTITYMenuAppendix

MATRICES: INVERSESMenuAppendix

MATRICES: SYSTEM OF EQUATIONSMenuAppendix

13

2 3 4 46

3 2 5

x y z

x y z

x y z

Solve this system using any method you choose:1 1 1 13

2 3 4 46

3 2 1 5

1 23R R 1 0 1 7

4 1 0 18

5 0 1 31

1 3R R

1 32R R

1 3R R

6 0 0 24

4 1 0 18

5 0 1 31

1 0 0 4

4 1 0 18

5 0 1 31

1 6R

1 24R R

1 0 0 4

0 1 0 2

0 0 1 11

1 35R R

1 0 0 4

0 1 0 2

0 0 1 11

REDUCED ROW-ECHELON FORM


2 2 5

3 2 2 19

6 2 7

x y z

x y z

x y z

Solving a System of Equations with Matrices

1 2 2 5

3 2 2 19

6 1 2 7

X Y Z

To solve this equation, we will take out everything that we don’t change.



1 2 2 5

3 2 2 19

6 1 2 7

We want to perform linear operations like adding or subtracting rows, or multiplying/dividing a whole equation by a number.

We will do this until our First three columns are 0’s and 1’s

1 0 0 ?

0 1 0 ?

0 0 1 ?

This is calledREDUCED ROW-ECHELON FORM



1 2 2 5

3 2 2 19

6 1 2 7

7 3 0 2

4 0 4 24

9 1 0 12

Add row 1 and row 3

Add row 1 and row 2

Add row 2 and row 3



7 3 0 2

4 0 4 24

9 1 0 12

Row 3 x3 plus row 1

7 3 0 2

4 0 4 24

34 0 0 34

9 1 0 1227 3 0 367 33 0 2

Row 3 x3



7 3 0 2

4 0 4 24

34 0 0 34

7 3 0 2

4 0 4 24

1 0 0 1

Divide the bottom row by 34



7 3 0 2

4 0 4 24

1 0 0 1

7 3 0 2

0 0 4 20

1 0 0 1

Bottom row x4, subtract row 3 from row 2

4 0 4 24

1 0 0 1

4 0 4 24

4 0 0 4Row 3 x4



7 3 0 2

0 0 4 20

1 0 0 1

7 3 0 2

0 0 1 5

1 0 0 1

Divide row 2 by 4



7 3 0 2

0 0 1 5

1 0 0 1

Bottom row x7, subtract row 3 from row 17 3 0 2

1 0 0 1

7 3 0 2

7 0 0 7

0 3 0 9

0 0 1 5

1 0 0 1

Row 3 x7



Divide the top row by three

0 3 0 9

0 0 1 5

1 0 0 1

1001

5100

3010



0 1 0 3

0 0 1 5

1 0 0 1

0 1 0 3x y z

0 0 1 5x y z

1 0 0 1x y z 3y

5z

1 x

5100

3010

1001X Y Z


MATRICES: APPENDIXMenuAppendix

OPENERS

ASSIGNMENTS

EXTRA PROBLEMS

USING A CALCULATOR

MATRICES: OPENERSMenuAppendix

ABCDEFGHIJKLMNOP

MATRICES: OPENERS OMenuAppendix

Solve this system of equations using any method you choose:

14

2 3 4 16

2 3 4 26

2 2 13

4 5 6 2 7

a b c d e

a b c

b a c d e

a b c d e

a b c d e

3

2

1

5

7

a

b

c

d

e

1 1 1 1 1 14

2 3 4 0 0 16

2 1 3 4 1 26

2 2 1 1 1 13

4 5 1 6 2 7

Gaussian EliminationReduced Row Echelon Form

MATRICES: ASSIGNMENTSMenuAppendix

p. 203p. 211p. 218Ch. 4 review booklet

MATRICES: EXTRA PROBLEMSMenuAppendix

CHAPTER 4 DIGITAL STATIONS

MATRICES: EXTRA PROBLEMSMenuAppendix

Basic Operations MultiplicationDeterminants Cramer’s RuleIdentityInverses Solving Systems of...

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