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# Solutions

## Solutions to Try Its

1. $A+B=\left[\begin{array}{c}2\\ 1\\ 1\end{array}\begin{array}{c}6\\ \text{ }\text{ }\text{ }0\\ -3\end{array}\right]+\left[\begin{array}{c}3\\ 1\\ -4\end{array}\begin{array}{c}-2\\ 5\\ 3\end{array}\right]=\left[\begin{array}{c}2+3\\ 1+1\\ 1+\left(-4\right)\end{array}\begin{array}{c}6+\left(-2\right)\\ 0+5\\ -3+3\end{array}\right]=\left[\begin{array}{c}5\\ 2\\ -3\end{array}\begin{array}{c}4\\ 5\\ 0\end{array}\right]$ 2. $-2B=\left[\begin{array}{cc}-8& -2\\ -6& -4\end{array}\right]$

## 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 $2\times 2$ matrix and the second is a $2\times 3$ matrix. $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]+\left[\begin{array}{ccc}6& 5& 4\\ 3& 2& 1\end{array}\right]$ has no sum. 3. Yes, if the dimensions of $A$ are $m\times n$ and the dimensions of $B$ are $n\times m,\text{}$ both products will be defined. 5. Not necessarily. To find $AB,\text{}$ we multiply the first row of $A$ by the first column of $B$ to get the first entry of $AB$. To find $BA,\text{}$ 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. $\left[\begin{array}{cc}11& 19\\ 15& 94\\ 17& 67\end{array}\right]$ 9. $\left[\begin{array}{cc}-4& 2\\ 8& 1\end{array}\right]$ 11. Undidentified; dimensions do not match 13. $\left[\begin{array}{cc}9& 27\\ 63& 36\\ 0& 192\end{array}\right]$ 15. $\left[\begin{array}{cccc}-64& -12& -28& -72\\ -360& -20& -12& -116\end{array}\right]$ 17. $\left[\begin{array}{ccc}1,800& 1,200& 1,300\\ 800& 1,400& 600\\ 700& 400& 2,100\end{array}\right]$ 19. $\left[\begin{array}{cc}20& 102\\ 28& 28\end{array}\right]$ 21. $\left[\begin{array}{ccc}60& 41& 2\\ -16& 120& -216\end{array}\right]$ 23. $\left[\begin{array}{ccc}-68& 24& 136\\ -54& -12& 64\\ -57& 30& 128\end{array}\right]$ 25. Undefined; dimensions do not match. 27. $\left[\begin{array}{ccc}-8& 41& -3\\ 40& -15& -14\\ 4& 27& 42\end{array}\right]$ 29. $\left[\begin{array}{ccc}-840& 650& -530\\ 330& 360& 250\\ -10& 900& 110\end{array}\right]$ 31. $\left[\begin{array}{cc}-350& 1,050\\ 350& 350\end{array}\right]$ 33. Undefined; inner dimensions do not match. 35. $\left[\begin{array}{cc}1,400& 700\\ -1,400& 700\end{array}\right]$ 37. $\left[\begin{array}{cc}332,500& 927,500\\ -227,500& 87,500\end{array}\right]$ 39. $\left[\begin{array}{cc}490,000& 0\\ 0& 490,000\end{array}\right]$ 41. $\left[\begin{array}{ccc}-2& 3& 4\\ -7& 9& -7\end{array}\right]$ 43. $\left[\begin{array}{ccc}-4& 29& 21\\ -27& -3& 1\end{array}\right]$ 45. $\left[\begin{array}{ccc}-3& -2& -2\\ -28& 59& 46\\ -4& 16& 7\end{array}\right]$ 47. $\left[\begin{array}{ccc}1& -18& -9\\ -198& 505& 369\\ -72& 126& 91\end{array}\right]$ 49. $\left[\begin{array}{cc}0& 1.6\\ 9& -1\end{array}\right]$ 51. $\left[\begin{array}{ccc}2& 24& -4.5\\ 12& 32& -9\\ -8& 64& 61\end{array}\right]$ 53. $\left[\begin{array}{ccc}0.5& 3& 0.5\\ 2& 1& 2\\ 10& 7& 10\end{array}\right]$ 55. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ 57. $\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ 59. ${B}^{n}=$ [latex-display]{B}^{n=\text{ even}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right][/latex-display] [latex-display]{B}^{n=\text{ odd}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right][/latex-display]