4.3 Matrix Multiplication

1. Multiplying a Matrix by a Scalar

2. Multiplying Matrices

1) Multiplying a Matrix by a Scalar

The number or factor you multiply a matrix by is called a scalar.

When multiplying by a scalar, every element in the matrix gets multiplied.

1) Multiplying a Matrix by a Scalar

Example 1:

Find 3-2 -3 4

0 9 3.4

3 is a scalar

1) Multiplying a Matrix by a Scalar

Example 1:

Find 3

=

-2 -3 4

0 9 3.4

-6 -9 12

0 27 10.2

1) Multiplying a Matrix by a Scalar

Example 2: Find X.

6 9

-12 15

27 -18

30 6=-3X + 2

1) Multiplying a Matrix by a Scalar

Example 2: Find X.

6 9

-12 15

27 -18

30 6=-3X + 2

12 18

-24 30

27 -18

30 6=-3X +

1) Multiplying a Matrix by a Scalar

Example 2: Find X.

6 9

-12 15

27 -18

30 6=-3X + 2

12 18

-24 30

27 -18

30 6=-3X +

12 18

-24 30

27 -18

30 6

=-3X -

1) Multiplying a Matrix by a Scalar

Example 2: Find X.

6 9

-12 15

27 -18

30 6=-3X + 2

12 18

-24 30

27 -18

30 6=-3X +

15 -36

54 -24

=-3X

1) Multiplying a Matrix by a Scalar

Example 2: Find X.

6 9

-12 15

27 -18

30 6=-3X + 2

12 18

-24 30

27 -18

30 6=-3X +

-5 12

-18 8

=X

2) Multiplying Matrices

Multiply the rows of the first matrix by the columns of the second matrix and add.

Stay organized or it will get messy…fast.

2) Multiplying Matrices

Example 1:

Find the product of and-3 3

5 0

-1 0

3 -4

2) Multiplying Matrices

Example 1:

Find the product of and

Answer: Some matrix with dimensions 2x2

-3 3

5 0

-1 0

3 -4

? ?

? ?

2) Multiplying Matrices

Example 1:

Find the product of and-3 3

5 0

-1 0

3 -4

Multiply row 1 and column 1

2) Multiplying Matrices

Example 1:

Find the product of and

= (-3)(-1) + 3(3)

= 12

-3 3

5 0

-1 0

3 -4

Multiply row 1 and column 1

2) Multiplying Matrices

Example 1:

Find the product of and

= (-3)(-1) + 3(3)

= 12

-3 3

5 0

-1 0

3 -4

Multiply row 1 and column 1

12 ?

? ?

2) Multiplying Matrices

Example 1:

Find the product of and

= (-3)(0) + 3(-4)

= -12

-3 3

5 0

-1 0

3 -4

Multiply row 1 and column 2

12 ?

? ?

2) Multiplying Matrices

Example 1:

Find the product of and

= (-3)(0) + 3(-4)

= -12

-3 3

5 0

-1 0

3 -4

Multiply row 1 and column 2

12 -12

? ?

2) Multiplying Matrices

Example 1:

Find the product of and

= (5)(-1) + 0(3)

= -5

-3 3

5 0

-1 0

3 -4

Multiply row 2 and column 1

12 -12

? ?

2) Multiplying Matrices

Example 1:

Find the product of and

= (5)(-1) + 0(3)

= -5

-3 3

5 0

-1 0

3 -4

Multiply row 2 and column 1

12 -12

-5 ?

2) Multiplying Matrices

Example 1:

Find the product of and

= (5)(0) + 0(-4)

= 0

-3 3

5 0

-1 0

3 -4

Multiply row 2 and column 2

12 -12

-5 ?

2) Multiplying Matrices

Example 1:

Find the product of and

= (5)(0) + 0(-4)

= 0

-3 3

5 0

-1 0

3 -4

Multiply row 2 and column 2

12 -12

-5 0

2) Multiplying Matrices

Example 2:

Find the product of and-3 3

5 0

-1 0

3 -4

2) Multiplying Matrices

Example 2:

Find the product of and-3 3

5 0

-1 0

3 -4

? ?

? ?

2) Multiplying Matrices

Example 2:

Find the product of and-3 3

5 0

-1 0

3 -4

? ?

? ?

2) Multiplying Matrices

Example 2:

Find the product of and

= (-1)(-3) + (0)(5)

= 3

-3 3

5 0

-1 0

3 -4

? ?

? ?

2) Multiplying Matrices

Example 2:

Find the product of and

= (-1)(-3) + (0)(5)

= 3

-3 3

5 0

-1 0

3 -4

3 ?

? ?

2) Multiplying Matrices

Example 2:

Find the product of and

= (-1)(3) + (0)(5)

= -3

-3 3

5 0

-1 0

3 -4

3 ?

? ?

2) Multiplying Matrices

Example 2:

Find the product of and

= (-1)(3) + (0)(5)

= -3

-3 3

5 0

-1 0

3 -4

3 -3

? ?

2) Multiplying Matrices

Example 2:

Find the product of and

= (3)(-3) + (-4)(5)

= -29

-3 3

5 0

-1 0

3 -4

3 -3

? ?

2) Multiplying Matrices

Example 2:

Find the product of and

= (3)(-3) + (-4)(5)

= -29

-3 3

5 0

-1 0

3 -4

3 -3

-29 ?

2) Multiplying Matrices

Example 2:

Find the product of and

= (3)(3) + (-4)(0)

= 9

-3 3

5 0

-1 0

3 -4

3 -3

-29 ?

2) Multiplying Matrices

Example 2:

Find the product of and

= (3)(3) + (-4)(0)

= 9

-3 3

5 0

-1 0

3 -4

3 -3

-29 9

4.2 Quiz Prep

p.180 #35-38

4.2 Quiz Prep Answers

p.180 #35-38

35) B

36) G

37) x = 4, y = -2, w = 0, z = 6

38) x = 5, y = 1

2) Multiplying Matrices

Example 3:

Find the product of A = and B =-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 3:

Find the product of A = and B =

**The matrices are different dimensions

-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 3:

Find the product of A = and B =

**The matrices are different dimensions

**How do you know if a matrix product exists?

-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 3:

Find the product of A = and B = -2 5 2

5 0.5 1

-3 7

4 2

AB BA

(2 x 2)(2 x 3) (2 x 3)(2 x 2)

2) Multiplying Matrices

Example 3:

Find the product of A = and B = -2 5 2

5 0.5 1

-3 7

4 2

AB BA

(2 x 2)(2 x 3) (2 x 3)(2 x 2)

2) Multiplying Matrices

Example 3:

Find the product of A = and B = -2 5 2

5 0.5 1

-3 7

4 2

AB BA

(2 x 2)(2 x 3) (2 x 3)(2 x 2)

2) Multiplying Matrices

Example 3:

Find the product of A = and B = -2 5 2

5 0.5 1

-3 7

4 2

AB BA

(2 x 2)(2 x 3) (2 x 3)(2 x 2)

Columns of A match rows of B

Columns of B do not match rows of A

2) Multiplying Matrices

Example 3:

Find the product of A = and B = -2 5 2

5 0.5 1

-3 7

4 2

AB BA

(2 x 2)(2 x 3) (2 x 3)(2 x 2)

Columns of A match rows of B

Columns of B do not match rows of A

Product AB is defined

Product BA is undefined

2) Multiplying Matrices

Example 3:

Find the product of A = and B =

Dimensions of the product matrix

= (2 x 2)(2 x 3)

-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 3:

Find the product of A = and B =

Dimensions of the product matrix

= (2 x 2)(2 x 3)

-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 3:

Find the product of A = and B =

Dimensions of the product matrix

= (2 x 2)(2 x 3)

= (2 x 3)

-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 3:

Find the product of A = and B =

-3(-2) + 7(5) -3(5) + 7(0.5) -3(2) + 7(1)

=

4(-2) + 2(5) 4(5) + 2(0.5) 4(2) + 2(1)

-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 3:

Find the product of A = and B =

41 -11.5 1

=

2 21 10

-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 4:

Find the product of B = and A =-2 5 2

5 0.5 1

-3 7

4 2

2) Multiplying Matrices

Example 4:

Find the product of B = and A =

= (2 x 3)(2 x 2)

= undefined

-2 5 2

5 0.5 1

-3 7

4 2

Homework

p.186 #5, 7, 10, 13-15, 19, 20-24, 40, 51

Tomorrow

In-class assignment

Friday

Quiz – section 4.3

Select Homework Solutions

p.186 #5, 7, 10, 13-15, 19, 20-24, 40, 51

14) [34] 23) Undefined15) [34 0] 24) Defined20) Defined 51) x = -3, y = -921) Defined22) Defined