Section 9.4 Matrix Operations Copyright ©2013, 2009, 2006, 2005 Pearson Education, Inc.

Objectives Add, subtract, and multiply matrices when possible. Write a matrix equation equivalent to a system of equations.

Matrices A capital letter is generally used to name a matrix, and lower-case letters with double subscripts generally denote its entries. For example, a23 read “a sub two three,” indicates the entry in the second row and the third column. Two matrices are equal if they have the same order and corresponding entries are equal.

Matrix Addition and Subtraction To add or subtract matrices, we add or subtract their corresponding entries. The matrices must have the same order. Addition and Subtraction of Matrices Given two m  n matrices A = [aij] and B = [bij], their sum is A + B = [aij + bij] and their difference is A  B = [aij  bij].

Example Find A + B for each of the following. a) b)

Example continued We have a pair of 2  2 matrices in part (a) and a pair of 3  2 matrices in part (b). Since each pair has the same order we can add their corresponding entries. a) b)

Examples Find C  D for each of the following. a) b)

Examples a) Since the order of each matrix is 3  2, we can subtract corresponding entries. b) Since the matrices do not have the same order, we cannot subtract them.

Scalar Multiplication When we find the product of a number and a matrix, we obtain a scalar product. The scalar product of a number k and a matrix A is the matrix denoted kA, obtained by multiplying each entry of A by the number k. The number k is called a scalar.

Example Find 4A and (2)A for

Properties of Matrix Addition and Scalar Multiplication For any m  n matrices, A, B, and C and any scalars k and l: Commutative Property of Addition A + B = B + A. Associative Property of Addition A + (B + C) = (A + B) + C. Associative Property of Scalar Multiplication (kl)A = k(lA).

More Properties Distributive Property k(A + B) = kA + kB. (k + l)A = kA + lA. Additive Identity Property There exists a unique matrix 0 such that: A + 0 = 0 + A = A. Additive Inverse Property There exists a unique matrix A such that: A + (A) = A + A = 0.

Matrix Multiplication For an m  n matrix A = [aij] and an n  p matrix B = [bij], the product AB = [cij] is an m  p matrix, where cij = ai1 • b1j + ai2 • b2j + ai3 • b3j + … + ain • bnj. We can multiply two matrices only when the number of columns in the first matrix is equal to the number of rows in the second matrix.

Example For find each of the following. a) AB b) BA c) AC

Solution AB A is a 2  3 matrix and B is a 3  2 matrix, so AB will be a 2  2 matrix.

Solution BA B is a 3  2 matrix and A is a 2  3 matrix, so BA will be a 3  3 matrix.

Solution AC The product AC is not defined because the number of columns of A, 3, is not equal to the number of rows of C, 2. Note that AB  BA. Multiplication of matrices is generally not commutative.

Application Dalton’s Dairy produces no-fat ice cream and frozen yogurt. The following table shows the number of gallons of each product that are sold at the dairy’s three retail outlets one week. On each gallon of no-fat ice cream, the dairy’s profit is $4, and on each gallon of frozen yogurt, it is $3. Use matrices to find the total profit on these items at each store for the given week. 120 80 Store 2 100 160 Frozen Yogurt (in gallons) No-fat Ice Cream (in gallons) Store 3 Store 1

Application continued We can write the table showing the distribution as a 2  3 matrix. The profit per gallon can also be written as a matrix. The total profit at each store is given by the matrix product PD.

Application continued The total profit on no-fat ice cream and frozen yogurt for the given week was $880 at store 1, $680 at store 2, and $780 at store 3.

Properties of Matrix Multiplication For matrices A, B, and C, assuming that the indicated operations are possible: Associative Property of Multiplication A(BC) = (AB)C. Distributive Property A(B + C) = AB + AC. (B + C)A = BA + CA.

Matrix Equations We can write a matrix equation equivalent to a system of equations. Example: Can be written as:

Matrix Equations If we let We can write this matrix equation as AX = B.