12.2 Multiplication of Matrices

Matrix Multiplication The product of two matrices, A m×p and B p×n, is the matrix AB with dimensions m × n. Any element in the ith row and jth column of this product matrix is the sum of the products of the corresponding elements of the ith row of A and the jth column of B. When you multiply matrices, they need to be conformable for multiplication. This means: # of columns in 1 st matrix = # of rows in 2 nd matrix Ex 1) 3 × 22 × 2 match dimensions of product 3 × 2 To get each element: this is the first row, first column so we take 1 st row of A × 1 st column of B (4)(2) + (5)(5) = = 33 write yourself how to get this element (4)(3) + (5)(6) = = 42 (7)(2) + (2)(5) = = 24

Ex 2) Find A 2 (same A from Ex 1) 3 × 2 Wait! You can’t! So… undefined We can solve for unknown elements in a matrix equation. Ex 3) Solve for x and y. 3x – 4 = 2 3x = 6 x = y = –3 5y = –15 y = –3

The Identity Matrix The identity matrix is the equivalent to the algebraic 1. Multiplying by it does not change the original. etc. Pattern: 1’s along the diagonal & 0’s everywhere else *If the product of two matrices is I, then they are inverses of each other. You can also multiply by a 0 matrix to get an O matrix.

Properties of Matrix Multiplication for Square Matrices If A, B, and C are n × n matrices, then AB is an n × n matrix.Closure (AB)C = A(BC)Associative I n×n A = AI n×n = AMultiplicative Identity O n×n A = AO n×n = O n×n Multiplicative Property of the Zero Matrix A –1 is the multiplicative inverse of A if A –1 is defined and AA –1 = A –1 A= I n×n Multiplicative Inverse A(B + C)= AB + AC (B + C)A = BA + CA Distributive Properties What properties are not here?? Commutative! When we “store” information in matrices, we may have to transpose them (switch rows & columns) to make them conformable for multiplication. Ex: Per 3 Per 4 Boys Girls It’s still the same!

Ex 4) A fruit stand owner packages fruit in three different ways for gift packages. Economy package, E, has 6 apples, 3 oranges and 3 pears. Standard package, S, has 5 apples, 4 oranges and 4 pears. Luxury package, L, has 6 types of each fruit. The costs are $0.50 for an apple, $1.10 for an orange and $0.80 for a pear. What is the total cost of preparing each package of fruit? Cost apple orange pear Number of Items apple orange pear If we multiply in this state… the labels don’t match up cost per fruitpackage per fruit cost per fruitfruit per package Package E S L 1 × 33 × 3 1 × 3

Homework #1202 Pg 608 #1, 3, 5, 8, 15, 16, 18, 19, 20, 29, 31, 34, 41, 42