578 CHAPTER 9 Mathematical Systems In Exercises 17–20, let A B C 1 2 0 5 , 3 2 5 0 , and 2 3 4 0 . = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ Determine the following. 17. B3 9 6 15 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 18. A2− 2 4 0 10 − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 19. B C 2 3 + 0 13 22 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 20. B C 3 2 − 13 0 7 0 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ In Exercises 21–26, determine A B. × 21. A B 1 3 0 6 , 2 6 8 4 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 26 18 48 24 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 22. A B 1 1 2 6 , 4 2 3 2 = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 7 0 10 16 − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 23. A B 2 3 1 0 4 6 , 2 4 1 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 15 22 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 24. A B 1 0 4 2 1 3 , 2 3 = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 2 14 11 − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 25. A [2 5 0], = − B 4 1 1 0 2 6 = − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ 13 2 ⎡ ⎣ ⎤ ⎦ 26. A B 3 2 , 2 5 0 0 7 1 = ⎡⎣ ⎤ ⎦ = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 6 1 2 ⎡ − ⎣ ⎤ ⎦ In Exercises 27–34, determine A B + and A B. × If an operation cannot be performed, explain why. 27. A B 2 5 0 4 , 1 0 6 3 = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 28. A B 2 3 6 7 , 1 3 3 4 = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 29. A B 1 3 0 2 4 1 , 7 2 3 2 1 1 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 30. A B 6 5 4 3 2 1 , 6 5 4 3 2 1 = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ * 31. A B 2 5 1 8 3 6 , 3 2 4 6 2 0 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ * 32. A B 1 2 3 4 , 3 2 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ Cannot be added; 1 1− ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 33. A B 3 4 , 5 9 8 = ⎡⎣ ⎤ ⎦ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ Cannot be added; cannot be multiplied 34. A B 6 4 1 2 3 4 , 1 0 4 1 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Cannot be added; cannot be multiplied In Exercises 35–38, show the commutative property of addition, A B B A, + = + holds for matrices A and B. 35. A B 3 1 1 4 , 4 1 7 3 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ A B B A 7 2 6 1 + = + = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ 36. A B 9 4 1 7 , 2 0 1 6 = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 37. A B 0 1 3 4 , 8 1 3 4 = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 38. A B 4 2 9 6 1 0 3 5 7 , 11 2 7 2 0 5 1 4 10 = − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − − − − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ * In Exercises 39– 42, show that the associative property of addition, A B C A B C ( ) ( ), + + = + + holds for matrices A B, , and C. 39. A B C 6 5 1 3 , 3 4 2 7 , 2 4 5 0 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ * 40. A B 7 4 9 36 , 5 6 1 4 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ C 7 5 1 3 = − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * 41. A B C 3 2 2 0 5 9 , 1 8 0 6 4 4 , 0 4 1 9 9 6 = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ * 42. A B 5 2 1 11 3 0 , 4 1 8 6 4 0 , = − − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ C 0 5 2 3 1 9 = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ * In Exercises 43– 46, determine whether A B B A × = × holds for matrices A and B. 43. A B 1 2 4 3 , 1 3 2 4 = − − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ − − ⎣ ⎢ ⎤ ⎦ ⎥ No 44. A B 4 2 1 3 , 2 4 3 1 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ No 45. A B 2 0 4 1 , 2 0 8 4 = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ = ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Yes 46. A B 2 1 3 3 , 3 1 3 2 = ⎡ − ⎣ ⎢ ⎤ ⎦ ⎥ = − ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ Yes *See Instructor Answer Appendix
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