1. Can we add any two matrices together? If so, explain why; if not, explain why not and give an example of two matrices that cannot be added together.
2. Can we multiply any column matrix by any row matrix? Explain why or why not.
3. Can both the products AB and BA be defined? If so, explain how; if not, explain why.
4. Can any two matrices of the same size be multiplied? If so, explain why, and if not, explain why not and give an example of two matrices of the same size that cannot be multiplied together.
5. Does matrix multiplication commute? That is, does AB=BA? If so, prove why it does. If not, explain why it does not.
For the following exercises, use the matrices below and perform the matrix addition or subtraction. Indicate if the operation is undefined.
18. AB
19. BC
20. CA
21. BD
22. DC
23. CB
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed.
24. A+B−C
25. 4A+5D
26. 2C+B
27. 3D+4E
28. C−0.5D
29. 100D−10E
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A2=A⋅A)
A=[−1052025],B=[40−201030],C=−1010−10
30. AB
31. BA
32. CA
33. BC
34. A2
35. B2
36. C2
37. B2A2
38. A2B2
39. (AB)2
40. (BA)2
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. (Hint: A2=A⋅A)
41. AB
42. BA
43. BD
44. DC
45. D2
46. A2
47. D3
48. (AB)C
49. A(BC)
For the following exercises, use the matrices below to perform the indicated operation if possible. If not possible, explain why the operation cannot be performed. Use a calculator to verify your solution.