2. Copyright © 2009 Pearson Education, Inc. CHAPTER 9: Systems of Equations and Matrices 9.1 Systems of Equations in Two Variables 9.2 Systems of Equations in Three Variables 9.3 Matrices and Systems of Equations 9.4 Matrix Operations 9.5 Inverses of Matrices 9.6 Determinants and Cramer’s Rule 9.7 Systems of Inequalities and Linear Programming 9.8 Partial Fractions

3. Copyright © 2009 Pearson Education, Inc. 9.4 Matrix Operations  Add, subtract, and multiply matrices when possible.  Write a matrix equation equivalent to a system of equations.

4. Slide 9.4-4 Copyright © 2009 Pearson Education, Inc. Matrices A capital letter is generally used to name a matrix, and lower- case letters with double subscripts generally denote its entries. For example, a 23 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.

5. Slide 9.4-5 Copyright © 2009 Pearson Education, Inc. 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 = [a ij ] and B = [b ij ], their sum is A + B = [a ij + b ij ] and their difference is A  B = [a ij  b ij ].

6. Slide 9.4-6 Copyright © 2009 Pearson Education, Inc. Example Find A + B for each of the following. a) b)

7. Slide 9.4-7 Copyright © 2009 Pearson Education, Inc. 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)

8. Slide 9.4-8 Copyright © 2009 Pearson Education, Inc. Examples Find C  D for each of the following. a) b)

9. Slide 9.4-9 Copyright © 2009 Pearson Education, Inc. 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.

10. Slide 9.4-10 Copyright © 2009 Pearson Education, Inc. 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.

11. Slide 9.4-11 Copyright © 2009 Pearson Education, Inc. Example Find 4A and (  2)A for. Solution:

12. Slide 9.4-12 Copyright © 2009 Pearson Education, Inc. 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).

13. Slide 9.4-13 Copyright © 2009 Pearson Education, Inc. 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.

14. Slide 9.4-14 Copyright © 2009 Pearson Education, Inc. Matrix Multiplication For an m  n matrix A = [a ij ] and an n  p matrix B = [b ij ], the product AB = [c ij ] is an m  p matrix, where c ij = a i1 b 1j + a i2 b 2j + a i3 b 3j + … + a in b nj. 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.

15. Slide 9.4-15 Copyright © 2009 Pearson Education, Inc. Examples For find each of the following. a) AB b) BA c) AC

16. Slide 9.4-16 Copyright © 2009 Pearson Education, Inc. Solution AB A is a 2  3 matrix and B is a 3  2 matrix, so AB will be a 2  2 matrix.

17. Slide 9.4-17 Copyright © 2009 Pearson Education, Inc. Solution BA B is a 3  2 matrix and A is a 2  3 matrix, so BA will be a 3  3 matrix.

18. Slide 9.4-18 Copyright © 2009 Pearson Education, Inc. 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.

19. Slide 9.4-19 Copyright © 2009 Pearson Education, Inc. 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 100160Frozen Yogurt (in gallons) 120100No-fat Ice Cream (in gallons) Store 3Store 1

20. Slide 9.4-20 Copyright © 2009 Pearson Education, Inc. 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.

21. Slide 9.4-21 Copyright © 2009 Pearson Education, Inc. 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.

22. Slide 9.4-22 Copyright © 2009 Pearson Education, Inc. 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.

23. Slide 9.4-23 Copyright © 2009 Pearson Education, Inc. Matrix Equations We can write a matrix equation equivalent to a system of equations. Example: Can be written as:

24. Slide 9.4-24 Copyright © 2009 Pearson Education, Inc. Matrix Equations If we let We can write this matrix equation as AX = B.