Lesson 12.2

Matrix Multiplication

3

Row and Column Order

The rows in a matrix are usually indexed 1 to m from top to bottom. The columns are usually indexed 1 to n from left to right. Elements are

indexed by row, then column.

nmmm

n

n

ji

aaa

aaa

aaa

a

,2,1,

,22,21,2

,12,11,1

, ][

A

If A is an m × n matrix and s is a scalar, then we let kA denote the matrix obtained by multiplying every element of A by k. This procedure is called scalar multiplication.

310

221A

930

663

331303

2323133A

In matrix algebra, a real number is often called a SCALAR. To multiply a matrix by a scalar, you multiply each entry in the matrix by that scalar.

14

024

Multiplying Matrices by a scalar

)1(4)4(4

)0(4)2(4

416

08

Scalar Multiplication - each element in a matrix is multiplied by a constant.

1. 2 7 1 0 -14 2 0

2. 4 2 0

4 5

-8 0

16 -20

3. 5

1 2 3

4 5 6

7 8 9

-5 -10 15

-20 25 -30

35 -40 45

86

54

30

212

)8(360

52412

-2

6

-3 3

-2(-3)

-5

-2(6) -2(-5)

-2(3) 6 -6

-12 10

Example

PEMDAS – parenthesis first, do the matrix addition

Do the scalar multiplication

The multiplication of matrices is easier shown than put into words. You multiply the rows of the first matrix with the columns of the second adding products

140

123A

13

31

42

B

Find AB

Multiply across the first row and down the first column adding products. Put the answer in the first row, first

column of the answer matrix.

23 1223 5311223

140

123A

13

31

42

B

Find AB

We multiplied across first row and down first column so we put the answer in the first row, first column.

5AB

Now we multiply across the first row and down the second column and we’ll put the answer in the first row, second column.

43 3243 7113243

75AB

Now we multiply across the second row and down the first column and we’ll put the answer in the second row, first column.

20 1420 1311420

1

75AB

Now we multiply across the second row and down the second column and we’ll put the answer in the second row,

second column.

40 3440 11113440

111

75AB

Notice the sizes of A and B and the size of the product AB.

Con’t

• You can multiply two matrices A and B only if the number of columns of A is equal to the number of rows of B.

Examples:

1. 2 1

3 4

3 9 2

5 7 6

2(3) + -1(5) 2(-9) + -1(7) 2(2) + -1(-6)

3(3) + 4(5) 3(-9) + 4(7) 3(2) + 4(-6)

1 25 10

29 1 18

3 9 2 2 1.

5 7 6 3 4

2

Dimensions: 2 x 3 2 x 2

*They don’t match so can’t be multiplied together.*

1 2 1 1

1 3 2 7

2 6 1 8

x

y

z

3.

x 2y z 1

x 3y 2z 7

2x 6y z 8

0 1 4 3

1 0 2 5

4.

2 x 2 2 x 2

*Answer should be a 2 x 2

0(4) + (-1)(-2) 0(-3) + (-1)(5)

1(4) + 0(-2) 1(-3) +0(5)

2 -5

4 -3