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