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.

k hA kh A

k h A kA hA

k A B kA kB

310

221A

930

663

331303

2323133A

PROPERTIES OF SCALAR MULTIPLICATION

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

The m × n zero matrix, denoted 0, is the m × n matrix whose elements are all zeros.

00

0)(

0

A

AA

AA

00

00 000

2 × 21 × 3

**Multiply rows times columns.**You can only multiply if the number of columns in the 1st matrix is equal to the number of rows in the 2nd matrix.

3 2 5

7 1 0

8 2

1 5

0 3

Dimensions: 3 x 22 x 3

They must match.

The dimensions of your answer.

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

First we multiply across the first row and down the first column adding products. We put the answer in the first row, first column of the answer.

23 1223 5311223

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

To multiply matrices A and B look at their dimensions

pnnm MUST BE SAME

SIZE OF PRODUCT

If the number of columns of A does not equal the number of rows of B then the

product AB is undefined.

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.

6

BA

126

BA

2126

BA

3

2126

BA

143

2126

BA

4143

2126

BA

9

4143

2126

BA

109

4143

2126

BA

4109

4143

2126

BA

Now let’s look at the product BA.

13

31

42

B

140

123A

BAAB

2332can multiply

size of answeracross first row as we go down first column:

60432

across first row as we go down second column:

124422

across first row as we go down third column:

21412

across second row as we go down first column:

30331

across second row as we go down second column:

144321

across second row as we go down third column:

41311

across third row as we go down first column:

90133

across third row as we go down second column:

104123

across third row as we go down third column:

41113

Completely different than AB!

Commuter's Beware!

BCACCBA

ACABCBA

CABBCA

PROPERTIES OF MATRIX MULTIPLICATION

BAAB Is it possible for AB = BA ?,yes it is possible.

Use

34

21A

34

25B

42

15C

And scalar c = 3 to determine whether the following equations are true for the given matrices.

a) AC + BC = (A+B)C

b) c(AB) = A(cB)

c) C(A + B) = AC + BC

d) ABC = CBA