Solutions to Try Its

1. [latex]A+B=\left[\begin{array}{c}2\\ 1\\ 1\end{array}\begin{array}{c}6\\ \text{ }\text{ }\text{ }0\\ -3\end{array}\right]+\left[\begin{array}{c}3\\ 1\\ -4\end{array}\begin{array}{c}-2\\ 5\\ 3\end{array}\right]=\left[\begin{array}{c}2+3\\ 1+1\\ 1+\left(-4\right)\end{array}\begin{array}{c}6+\left(-2\right)\\ 0+5\\ -3+3\end{array}\right]=\left[\begin{array}{c}5\\ 2\\ -3\end{array}\begin{array}{c}4\\ 5\\ 0\end{array}\right][/latex]

2. [latex]-2B=\left[\begin{array}{cc}-8& -2\\ -6& -4\end{array}\right][/latex]

Solutions to Odd-Numbered Exercises

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 [latex]2\times 2[/latex] matrix and the second is a [latex]2\times 3[/latex] matrix. [latex]\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ccc}6& 5& 4\\ 3& 2& 1\end{array}\right][/latex] has no sum.

3. Yes, if the dimensions of [latex]A[/latex] are [latex]m\times n[/latex] and the dimensions of [latex]B[/latex] are [latex]n\times m,\text{}[/latex] both products will be defined.

5. Not necessarily. To find [latex]AB,\text{}[/latex] we multiply the first row of [latex]A[/latex] by the first column of [latex]B[/latex] to get the first entry of [latex]AB[/latex]. To find [latex]BA,\text{}[/latex] we multiply the first row of [latex]B[/latex] by the first column of [latex]A[/latex] to get the first entry of [latex]BA[/latex]. Thus, if those are unequal, then the matrix multiplication does not commute.

7. [latex]\left[\begin{array}{cc}11& 19\\ 15& 94\\ 17& 67\end{array}\right][/latex]

9. [latex]\left[\begin{array}{cc}-4& 2\\ 8& 1\end{array}\right][/latex]

11. Undidentified; dimensions do not match

13. [latex]\left[\begin{array}{cc}9& 27\\ 63& 36\\ 0& 192\end{array}\right][/latex]

15. [latex]\left[\begin{array}{cccc}-64& -12& -28& -72\\ -360& -20& -12& -116\end{array}\right][/latex]

17. [latex]\left[\begin{array}{ccc}1,800& 1,200& 1,300\\ 800& 1,400& 600\\ 700& 400& 2,100\end{array}\right][/latex]

19. [latex]\left[\begin{array}{cc}20& 102\\ 28& 28\end{array}\right][/latex]

21. [latex]\left[\begin{array}{ccc}60& 41& 2\\ -16& 120& -216\end{array}\right][/latex]

23. [latex]\left[\begin{array}{ccc}-68& 24& 136\\ -54& -12& 64\\ -57& 30& 128\end{array}\right][/latex]

25. Undefined; dimensions do not match.

27. [latex]\left[\begin{array}{ccc}-8& 41& -3\\ 40& -15& -14\\ 4& 27& 42\end{array}\right][/latex]

29. [latex]\left[\begin{array}{ccc}-840& 650& -530\\ 330& 360& 250\\ -10& 900& 110\end{array}\right][/latex]

31. [latex]\left[\begin{array}{cc}-350& 1,050\\ 350& 350\end{array}\right][/latex]

33. Undefined; inner dimensions do not match.

35. [latex]\left[\begin{array}{cc}1,400& 700\\ -1,400& 700\end{array}\right][/latex]

37. [latex]\left[\begin{array}{cc}332,500& 927,500\\ -227,500& 87,500\end{array}\right][/latex]

39. [latex]\left[\begin{array}{cc}490,000& 0\\ 0& 490,000\end{array}\right][/latex]

41. [latex]\left[\begin{array}{ccc}-2& 3& 4\\ -7& 9& -7\end{array}\right][/latex]

43. [latex]\left[\begin{array}{ccc}-4& 29& 21\\ -27& -3& 1\end{array}\right][/latex]

45. [latex]\left[\begin{array}{ccc}-3& -2& -2\\ -28& 59& 46\\ -4& 16& 7\end{array}\right][/latex]

47. [latex]\left[\begin{array}{ccc}1& -18& -9\\ -198& 505& 369\\ -72& 126& 91\end{array}\right][/latex]

49. [latex]\left[\begin{array}{cc}0& 1.6\\ 9& -1\end{array}\right][/latex]

51. [latex]\left[\begin{array}{ccc}2& 24& -4.5\\ 12& 32& -9\\ -8& 64& 61\end{array}\right][/latex]

53. [latex]\left[\begin{array}{ccc}0.5& 3& 0.5\\ 2& 1& 2\\ 10& 7& 10\end{array}\right][/latex]

55. [latex]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right][/latex]

57. [latex]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right][/latex]

59. [latex]{B}^{n}=[/latex]

[latex]{B}^{n=\text{ even}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right][/latex]

[latex]{B}^{n=\text{ odd}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right][/latex]