1. If A−1 is the inverse of A, then AA−1=I, the identity matrix. Since A is also the inverse of A−1,A−1A=I. You can also check by proving this for a 2×2 matrix.
3. No, because ad and bc are both 0, so ad−bc=0, which requires us to divide by 0 in the formula.
5. Yes. Consider the matrix [0110]. The inverse is found with the following calculation: A−1=0(0)−1(1)1[0−1−10]=[0110].
7. AB=BA=[1001]=I
9. AB=BA=[1001]=I
11. AB=BA=100010001=I
13. 291[9−123]
15. 691[−2973]
17. There is no inverse
19. 74[0.511.5−0.5]
21. 171−52015−3−1−3124
23. 20914710−24−57193869−12−13
25. 18−564060−14080−168448−280
27. (−5,6)
29. (2,0)
31. (31,−25)
33. (−32,−611)
35. (7,21,51)
37. (5,0,−1)
39. 341(−35,−97,−154)
41. 6901(65,−1136,−229)
43. (−3037,158)
45. (12310,−1,52)
47. 21200011−11−111−1−1−111
49. 39131824−92−53−364613221−16−7109−5
51. 10000−101000−100100−100010−100001−1000001
53. Infinite solutions.
55. 50% oranges, 25% bananas, 20% apples
57. 10 straw hats, 50 beanies, 40 cowboy hats
59. Tom ate 6, Joe ate 3, and Albert ate 3.
61. 124 oranges, 10 lemons, 8 pomegranates