# Solutions 31: Matrices and Matrix Operations

## 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.
$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right] \text{even}\\ \left[\begin{array}{ccc}1& 0& 0\\ 0& 0& 1\\ 0& 1& 0\end{array}\right] \text{odd}$