3.6 Multiply Matrices

Example 1

Describe matrix products

State whether the product AB is defined. If so, give the dimensions of AB.

a. A: 2 ´ 3, B: 4 ´ 3 b. A: 3 ´ 3, B: 3 ´ 2

a. Because the number of __columns__ in A (three) __does not equal__ the number of __rows__ in B (four), the product AB __is not__ defined.

b. Because A is a 3 ´ 3 matrix and B is a 3 ´ 2 matrix, the product AB __is__ defined and is a __3 ´ 2__ matrix.

MULTIPLYING MATRICES

Words To find the element in the ith row and jth column of the product matrix AB, multiply each element in the __ith row of A__ by the corresponding element in the __jth column of B__, then add the products.

A B AB

Algebra

Example 2

Find the product of two matrices

Find AB if and .

Because A is a 2 ´ 2 matrix and B is a 2 ´ 2 matrix, the product AB __is__ defined and is a __2 ´ 2__ matrix.

Given A and B, give the dimensions of AB. Then find AB.

1.

2.

Example 3

Use matrix operations

If and evaluate each expression.

a. (A + B)C b. AC + BC

Solution

a. (A + B) C

b. AC + BC

PROPERTIES OF MATRIX MULTIPLICATION

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

Associative Property of Matrix Multiplication A(BC) = __(AB)C__

Left Distributive Property A(B + C) = __AB + AC__

Right Distributive Property (A + B)C = __AC + BC__

Associative Property of Scalar Multiplication k(AB) = __(kA)B_ = __A(kB)__

Example 4

Use matrices to calculate total cost

The school stores from the middle school and the high school each submit an inventory list for the year. Each sweatshirt costs $15, each T-shirt costs $9, and each pennant costs $5. Use matrix multiplication to find the total cost of the inventory for each school store.

Middle School: 61 sweatshirts, 63 T-shirts, and 74 pennants

High School: 58 sweatshirts, 71 T-shirts, and 92 pennants

Write the inventory and the cost in matrix form.

	Inventory		Cost Dollars
	Sweatshirt T-shirts Pennant	Sweatshirt
Middle High		T-shirt
		Pennant

Remember to set up the matrices so that the columns of the inventory matrix match the rows of the cost matrix.

Find the total cost of inventory for each school store by multiplying the inventory matrix by the cost matrix.

Label the product matrix: Total Cost Dollars

Middle School

High School

The total cost for the Middle School store is $ _1852_, and the total cost for the High School store is $ _1969_.