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Mathematics Question on Matrices

If A=\begin{bmatrix}1&2&-3\\\ 5&0&2\\\ 1&-1&1\end{bmatrix}$$,B=\begin{bmatrix}3&-1&2\\\ 4&2&5\\\ 2&0&3\end{bmatrix}$$,and\space C=\begin{bmatrix}4&1&2\\\ 0&3&2\\\ 1&-2&3\end{bmatrix}, then compute(A+B)and(B-C). Also, verify that A+(B-C)=(A+B)-C.

Answer

A+B=[12−3 502 1−11]+[3−12 425 203]A+B=\begin{bmatrix}1&2&-3\\\ 5&0&2\\\ 1&-1&1\end{bmatrix}+\begin{bmatrix}3&-1&2\\\ 4&2&5\\\ 2&0&3\end{bmatrix}
=[1+32−1−3+2 5+40+22+5 1+2−1+01+3]=\begin{bmatrix}1+3& 2-1& -3+2\\\ 5+4& 0+2& 2+5\\\ 1+2& -1+0& 1+3\end{bmatrix}
=[41−1 927 3−14]=\begin{bmatrix}4&1&-1\\\ 9&2&7\\\ 3&-1&4\end{bmatrix}
B−C=[3−12 425 203]−[412 032 1−23]B-C=\begin{bmatrix}3&-1&2\\\ 4&2&5\\\ 2&0&3\end{bmatrix}-\begin{bmatrix}4&1&2\\\ 0&3&2\\\ 1&-2&3\end{bmatrix}
=[3−4−1−12−2 4−02−35−2 2−10−(−2)3−3]=\begin{bmatrix}3-4& -1-1& 2-2\\\ 4-0& 2-3& 5-2\\\ 2-1& 0-(-2)& 3-3\end{bmatrix}
=[−1−20 4−13 120]=\begin{bmatrix}-1&-2&0\\\ 4&-1&3\\\ 1&2&0\end{bmatrix}
A+(B−C)=[12−3 502 1−11]+[−1−20\4−13\120]A+(B-C)=\begin{bmatrix}1&2&-3\\\ 5&0&2\\\ 1&-1&1\end{bmatrix}+\begin{bmatrix}-1&-2&0\\\4&-1&3\\\1&2&0\end{bmatrix}
=[1+(−1)2+(−2)−3+0 5+40+(−1)2+3 1+1−1+21+0]=\begin{bmatrix}1+(-1)& 2+(-2)& -3+0\\\ 5+4& 0+(-1)& 2+3\\\ 1+1& -1+2& 1+0\end{bmatrix}
=[00−3 9−15 211]=\begin{bmatrix}0&0&-3\\\ 9&-1&5\\\ 2&1&1\end{bmatrix}
(A+B)−C=[41−1 927 3−14]−[412 032 1−23](A+B)-C=\begin{bmatrix}4&1&-1\\\ 9&2&7\\\ 3&-1&4\end{bmatrix}-\begin{bmatrix}4&1&2\\\ 0&3&2\\\ 1&-2&3\end{bmatrix}
=[4−41−1−1−2 9−42−37−2 3−1−1−(−2)4−3]=\begin{bmatrix}4-4& 1-1& -1&-2\\\ 9-4& 2-3& 7-2\\\ 3-1& -1-(-2)& 4-3\end{bmatrix}
=[00−3 9−15 211]=\begin{bmatrix}0&0&-3\\\ 9&-1&5\\\ 2&1&1\end{bmatrix}
Hence,we have verfied that A+(B-C)=(A+B)-C.