Chapter 4 Section 2: Multiplying Matrices

VOCABULARY 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. IMPORTANT: Remember a matrix is written as ROW x COLUMN A is a 3x2 and B is a 2x3

If A is an m x n matrix and B is an n x p matrix, then the product of AB is an m x p matrix. It does not matter what m and p are as long as both ns are the same. m x n n x p m x p Dimensions of AB EQUAL A  B = AB

DESCRIBING MATRICES State whether the product of AB is defined. Is so, give the dimensions of AB. 1. A: 2 x 3, B: 3 x 4 2. A: 3 x 2, B: 3 x 4 3. A: 4 x 2, B: 2 x 3 Yes, 2 x 4 No Yes, 4 x 3

MULTIPLYING MATRICES

Multiplying Matrices Example 2

PROPERTIES OF MATRIX MULTIPLICATION Let A, B, and C be matrices with the same dimensions, 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 ASSOCATIVE PROPERTY OF SCALAR MULTIPLICATION k(AB) = (kA)B = A(kB)

Matrix multiplication is useful in business applications application because an inventory matrix, when multiplied by a cost per item matrix, results in a total cost matrix.

Two softball teams submit equipment lists for the season m x n n x p m x p Women’s Team 12 bats 45 balls 15 uniforms Men’s Team 15 bats 38 balls 17 uniforms Each bat cost $21, each ball costs $4, and each uniform costs $30. Use matrix multiplication to find the total cost of equipment for each team.