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MULTIPLYING TWO MATRICES

The product of two matrices A and B is defined

provided the number of columns in A is equal to

the number of rows in B.

MULTIPLYING TWO MATRICES

A ? B

? AB

4 X 3 ? 3 X 5 ?

4 X 5

4 rows

3 rows

3 columns

5 columns

MULTIPLYING TWO MATRICES

A ? B

? AB

4 X 3 ? 3 X 5 ?

4 X 5

4 rows

5 columns

MULTIPLYING TWO MATRICES

If A is a 4 X 3 matrix and B is a 3 X 5 matrix,

then the product AB is a 4 X 5 matrix.

MULTIPLYING TWO MATRICES

A ? B

? AB

m X n ? n X p ?

m X p

m rows

n rows

n columns

p columns

MULTIPLYING TWO MATRICES

A ? B

? AB

m X n ? n X p ?

m X p

m rows

p columns

MULTIPLYING TWO MATRICES

If A is an m X n matrix and B is an n X p

matrix, then the product AB is an m X p matrix.

Find AB if

A

and B

SOLUTION

Because A is a 3 X 2 matrix and B is a 2 X 2

matrix, the product AB is defined and is a 3 X 2

matrix.

To write the entry in the first row and first

column of AB, multiply corresponding entries in

the first row of A and the first column of B.

Then add.

Use a similar procedure to write the other

entries of the product.

A ? B ?

AB

3 X 2 ? 2 X 2 ?

3 X 2

A ? B ?

AB

3 X 2 ? 2 X 2 ?

3 X 2

A ? B ?

AB

3 X 2 ? 2 X 2 ?

3 X 2

A ? B ?

AB

3 X 2 ? 2 X 2 ?

3 X 2

?

MULTIPLYING TWO MATRICES

Matrix multiplication is not, in general,

commutative.

Let A, B, and C be matrices and let g be a scalar.

ASSOCIATIVE PROPERTY OF MATRIX MULTIPLICATION

A(BC ) (AB )C

A(B C ) AB AC

LEFT DISTRIBUTIVE PROPERTY

(A B )C AC BC

RIGHT DISTRIBUTIVE PROPERTY

ASSOCIATIVE PROPERTY OF SCALAR MULTIPLICATION

g (AB ) (gA )B A(gB )

USING MATRIX MULTIPLICATION IN REAL LIFE

Matrix multiplication is useful in business

applications because an inventory matrix, when

multiplied by a cost per item matrix, results in

total cost matrix.

m X n

n X p

m X p

For the total cost matrix to be meaningful, the

column labels for the inventory matrix must match

the row labels for the cost per item matrix.

SPORTS Two softball teams submit equipment lists

for the season.

Each bat costs 21, each ball costs 4, and each

uniform costs 30.

Use matrix multiplication to find the total cost

of equipment for each team.

SOLUTION

Write the equipment lists and costs per item in

matrix form.Use matrix multiplication to find

the total cost. Set up matrices so that columns

of the equipment matrix match rows of the cost

matrix.

COST

EQUIPMENT

Dollars

Balls

Uniforms

Bats

21

Womens team

12

45

15

Bats

4

15

38

17

Mens team

Balls

30

Uniforms

Total equipment cost for each team can be

obtained by multiplying the equipment matrix by

the cost per item matrix. The equipment matrix

is 2 X 3 and the cost per item matrix is 3 X

1. Their product is a 2 X 1 matrix.

12(21) 45(4) 15(30)

882

15(21) 38(4) 17(30)

977

TOTAL COST

Dollars

The labels of the product are

882

Womens team

977

Mens team

The total cost of equipment for the womens team

is 882,and the total cost of equipment for the

mens team is 977.