Question

If A = $\left[\begin{array}{ccc}1& -3& 2\\ 2& 0& 2\end{array}\right]$ and, B = $\left[\begin{array}{ccc}2& -1& -1\\ 1& 0& -1\end{array}\right]$, find the matrix C such that A + B + C is zero matrix.

Answer 1

$$\begin{gathered} \mathrm{Given}:\hspace{0.17em}A+B+C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}1& -3& 2\\ 2& 0& 2\end{array}\right]+\left[\begin{array}{ccc}2& -1& -1\\ 1& 0& -1\end{array}\right]+C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}1+2& -3-1& 2-1\\ 2+1& 0+0& 2-1\end{array}\right]+C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right] \\ \Rightarrow \left[\begin{array}{ccc}3& -4& 1\\ 3& 0& 1\end{array}\right]+C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right] \\ \Rightarrow C=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\end{array}\right]-\left[\begin{array}{ccc}3& -4& 1\\ 3& 0& 1\end{array}\right] \\ \Rightarrow C=\left[\begin{array}{ccc}0-3& 0+4& 0-1\\ 0-3& 0-0& 0-1\end{array}\right] \\ \Rightarrow C=\left[\begin{array}{ccc}-3& 4& -1\\ -3& 0& -1\end{array}\right] \end{gathered}$$