Question

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 1

\(A+B=\begin{bmatrix}1&2&-3\\ 5&0&2\\ 1&-1&1\end{bmatrix}+\begin{bmatrix}3&-1&2\\ 4&2&5\\ 2&0&3\end{bmatrix}\)
\(=\begin{bmatrix}1+3& 2-1& -3+2\\ 5+4& 0+2& 2+5\\ 1+2& -1+0& 1+3\end{bmatrix}\)
\(=\begin{bmatrix}4&1&-1\\ 9&2&7\\ 3&-1&4\end{bmatrix}\)
\(B-C=\begin{bmatrix}3&-1&2\\ 4&2&5\\ 2&0&3\end{bmatrix}-\begin{bmatrix}4&1&2\\ 0&3&2\\ 1&-2&3\end{bmatrix}\)
\(=\begin{bmatrix}3-4& -1-1& 2-2\\ 4-0& 2-3& 5-2\\ 2-1& 0-(-2)& 3-3\end{bmatrix}\)
\(=\begin{bmatrix}-1&-2&0\\ 4&-1&3\\ 1&2&0\end{bmatrix}\)
\(A+(B-C)=\begin{bmatrix}1&2&-3\\ 5&0&2\\ 1&-1&1\end{bmatrix}+\begin{bmatrix}-1&-2&0\\4&-1&3\\1&2&0\end{bmatrix}\)
\(=\begin{bmatrix}1+(-1)& 2+(-2)& -3+0\\ 5+4& 0+(-1)& 2+3\\ 1+1& -1+2& 1+0\end{bmatrix}\)
\(=\begin{bmatrix}0&0&-3\\ 9&-1&5\\ 2&1&1\end{bmatrix}\)
\((A+B)-C=\begin{bmatrix}4&1&-1\\ 9&2&7\\ 3&-1&4\end{bmatrix}-\begin{bmatrix}4&1&2\\ 0&3&2\\ 1&-2&3\end{bmatrix}\)
\(=\begin{bmatrix}4-4& 1-1& -1&-2\\ 9-4& 2-3& 7-2\\ 3-1& -1-(-2)& 4-3\end{bmatrix}\)
\(=\begin{bmatrix}0&0&-3\\ 9&-1&5\\ 2&1&1\end{bmatrix}\)
Hence,we have verfied that A+(B-C)=(A+B)-C.