Notes 7.2 – Matrices

I. Matrices A.) Def. – A rectangular array of numbers. An m

x n matrix is a matrix consisting of m rows and n columns. The element aij is the element found in row i, column j.

B.) Matrices are only considered to be equivalent when they have the same order, and their corresponding elements are equal.

C.) Matrix Addition and Subtraction

1.) Let A = [aij] and B = [bij].

Then, A + B = [aij + bij] and A – B = [aij - bij]

D.) Scalar Multiplication and Properties-

1.) Let A = [aij].

Then, kA = k[aij] = [kaij] where k is a constant

E.) Additive Identity, Zero Matrix, and Additive Inverse – See page 579

II. Basic Matrix Operations

A.) Ex. 1 - Let A = [aij] and B = [bij] be 3 x 3 matrices with aij = 3i – j and bij = i2+ j2.

1.) Find A and B.

2 1 0

5 4 3

8 7 6

A

2 5 10

5 8 13

10 13 18

B

2.) Find 3A+ 2B.

8 13 20

3 2 25 28 35

44 47 54

A B

2 1 0

5 4 3

8 7 6

A

1 5 10

5 8 13

10 13 18

B

Then A x B = an m by n matrix where

[aij] x [bij] = [cij] =

III. Matrix Multiplication

1 1 2 2 3 3 ...i j i j i j ir rja b a b a b a b

A.) Let A be an m by r matrix where A = [aij] and B be an r by n matrix where B = [bij].

B.) Ex. 2 - Find the product of A and B.

2 4 1

3 2 6A

1 0 2

0 1 4

4 3 0

B

2 7 12

21 20 2A B

2( 1) 4(0) 1(4) 2(0) 4(1) 1(3) 2(2) 4( 4) 1(0)

3( 1) 2(0) 6(4) 3(0) 2(1) 6(3) 3(2) 2( 4) 6(0)A B

C.) Ex. 3 - Find the product of A and B.

2 3

4 2

1 6

A

1 0 2

0 1 4

4 3 0

B

Note, a 3 x 2 cannot be multiplied by a 3 x 3 matrix. However, we can transpose matrix A to be a 2 by 3 matrix and then multiply it by B. This product is represented by ATB.

2 4 1

3 2 6TA

1 0 2

0 1 4

4 3 0

B

2 7 12

21 20 2TA B

AIn = InA = A - In is the multiplicative identity

IV. Identity Matrix In

3 3

1 0 0

0 1 0

0 0 1

I

A.) Def. – An n by n matrix with a main diagonal of all 1’s and zeros everywhere else.

B.) For real numbers, means that

is the multiplicative inverse of a.

A-1 is the multiplicative inverse of A.

11a

a

1

a

1 1NOTE: A

A

V. Finding the Inverse Matrix A-1

3 2

1 1A

A.) Ex. 4 – Find the multiplicative inverse of

3 2 1 0

1 1 0 1

a b

c d

3 2 1

3 2 0

0

1

a c

b d

a c

b d

3 2 1

3 0

1

1

a c

a c

c

a

3 2 1 2 1 0

1 1 1 3 0 1

1 2 is the inverse of

1 3A

3 2 0

3 1

3

2

b d

b d

d

b

There has to be an easier way to find the inverse, doesn’t there?!?

The answer is YES!!!!

VI. Determinants and Inverses (2 by 2)

1

1 1a b d bA

c d c aad bc

A.) Def. – If ad – bc ≠ 0, then

The number ad – bc is the determinant of a 2 by 2 matrix

5 4

2 3A

B.) Ex. 5– Find the multiplicative inverse of the following matrices

1

3 43 4 3 41 1 7 7

2 5 2 5 2 55( 3) 4( 2) 7

7 7

A or

4 1

8 2B

Therefore, B has no inverse. It is called a singular matrix.

1 1 1

4( 2) 8( 1) 0B

VII. Cofactors

1i j

ij ijA M

A.) Def. – Let A = [aij ] be a matrix of order n x n where n > 2.

Mij is the MINOR determinant.

(i.e., the determinant of the (n – 1) by (n – 1) matrix obtained by deleting the row and column containing aij .)

The cofactor corresponding to aij is

2 1 1

1 0 2

1 3 1

A

B.) Ex. 5– Find the all the cofactors for matrix A.

11 6A 1 1

11

0 21

3 1A

1 2

12

1 21 1

1 1A

1 3

13

1 01 3

1 3A

2 1 1

1 0 2

1 3 1

A

2 2

22

2 11 3

1 1A

2 3

23

2 11 4

1 3A

2 1

21

1 11 4

3 1A

3 2

32

2 11 5

1 2A

3 3

33

2 11 1

1 0A

3 1

31

1 11 2

0 2A

VII. Determinants and Inverses (n by n)

1 1 2 2det ...i i i i in inA A a A a A a A

A.) Thm – An n by n matrix A has an inverse iff det A ≠ 0.

B.) Def. – Let A = [aij ] be a matrix of order n x n where n > 2. The determinant of A, |A|, is the sum of the entries in any row or any column multiplied by their respective cofactors. For example, expanding by the ith row gives

2 1 1

1 0 2

1 3 1

A

2( 6) 1(1) 1( 3)A

14A

1(1) 0( 3) 3( 5)A

Using the first row

Using the second column

B.) Ex. 6 – Determine whether matrix A has an inverse by finding the determinant.

Yes, A has an inverse!

2 1 1

1 0 2

1 3 1

A

2 1 1 2 1

1 0 2 1 0

1 3 1 1 3

A

20

Rewrite the first 2 columns outside the matrix

Find each diagonal product and add them

C.) Another way to determine the determinant of a 3 x 3 matrix is to do the following

3 1

1 13 14A

2 1 1 2 1

1 0 2 1 0

1 3 1 1 3

A

120

Find each of the three opposite diagonal products and add them

Subtract your second product from the first and you have your determinant.

1 13

2 1 1

1 0 2

1 3 1

A

3 2 1

7 7 71 3 5

14 14 143 5 1

14 14 14

A

D.) Ex. – Find the inverse of A.

Matrix – Names - x-1 - ENTER

C.) The only way we have of finding the inverse of an n by n matrix for n > 3 is to use cofactors or our graphing calculator!

Matrix – Edit - Quit

VIII. Apps: Reflection Matrices

1 0

0 1

To reflect across the… multiply [x, y] by…

x-axis:

y-axis:

origin:

1 0

0 1

1 0

0 1