MATRICES MATRIX MULTIPLICATION Matrix Multiplication n n n

MATRICES MATRIX MULTIPLICATION

Matrix Multiplication n n n If A, B, and C are matrices and k is an integer, then matrix multiplication is: Associative: A(BC) = (AB)C Left Distributive: A(B+C) = AB+BC Right Distributive: (A+B)C=AC +BC Associative with Scalar Multiplication: k(AB) = (k. A)B = A(k. B) It is not commutative: AB ≠ BA

Matrix Multiplication § You can multiply matrices only if the number of columns in the first matrix equals the number of rows in the second matrix. 2 columns 2 rows

Matrix Multiplication § Notice the dimensions of the matrices and their product. 3 x 2 __ § 2 x__ 3 3 x__ 3 __ Note that if it had been a 2 x 3● 3 x 2, the result would have been a 2 x 2 matrix. Commutative does not work!

State Whether the Product is Defined for Matrices A and B n n n A: 3 X 6, B: 6 x 2 AB is a 3 x 2 matrix and is defined! A: 2 X 7, B: 1 X 7 AB is not defined! A: 4 X 2, B: 2 X 5 AB is a 4 x 5 matrix and is defined!

Matrix Multiplication § § So how do we multiply a matrix? With a math note sheet telling us how! Please write this down!

Example n

Matrix Multiplication n

Matrix Multiplication § Another example: 3 x 2 2 x 1 3 x 1

Example n

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Real Life Problem---Soccer Two soccer teams submit equipment lists for the season as shown: JV Varsity Balls Quick Goals Uniforms Costs Balls Quick Goals Uniforms How much will the total equipment cost for each team?

Real Life Problem---Soccer n