1. No, they must have the same dimensions. An example would include two matrices of different dimensions. One cannot add the following two matrices because the first is a 2×2 matrix and the second is a 2×3 matrix. [1324]+[635241] has no sum.
3. Yes, if the dimensions of A are m×n and the dimensions of B are n×m, both products will be defined.
5. Not necessarily. To find AB, we multiply the first row of A by the first column of B to get the first entry of AB. To find BA, we multiply the first row of B by the first column of A to get the first entry of BA. Thus, if those are unequal, then the matrix multiplication does not commute.
7. 111517199467
9. [−4821]
11. Undidentified; dimensions do not match
13. 96302736192
15. [−64−360−12−20−28−12−72−116]
17. 1,8008007001,2001,4004001,3006002,100
19. [202810228]
21. [60−16411202−216]
23. −68−54−5724−123013664128
25. Undefined; dimensions do not match.
27. −840441−1527−3−1442
29. −840330−10650360900−530250110
31. [−3503501,050350]
33. Undefined; inner dimensions do not match.
35. [1,400−1,400700700]
37. [332,500−227,500927,50087,500]
39. [490,00000490,000]
41. [−2−7394−7]
43. [−4−2729−3211]
45. −3−28−4−25916−2467
47. 1−198−72−18505126−936991
49. [091.6−1]
51. 212−8243264−4.5−961
53. 0.52103170.5210
55. 100010001
57. 100010001
59. {B}^{n}=\left\{\begin{array}{l}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right],\text{ }n\text{even,}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right],\text{ }n\text{odd}\text{.}\end{array}