If $\mathrm{A}=\left[\begin{array}{rrr}1 & -3 & 2 \\ 2 & 0 & 2\end{array}\right]$ and $\mathrm{B}=\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1\end{array}\right]$, find a matrix $\mathrm{C}$ such that $(\mathrm{A}+\mathrm{B}+\mathrm{C})$ is a zero matrix.
Given A+B+C is zero matrix i.e A+B+C = 0
$\left[\begin{array}{ccc}1 & -3 & 2 \\ 2 & 0 & 2\end{array}\right]+\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1\end{array}\right]+C=0$
$C=-\left[\begin{array}{ccc}1 & -3 & 2 \\ 2 & 0 & 2\end{array}\right]-\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1\end{array}\right]$
$=\left[\begin{array}{rrr}-3 & 4 & -1 \\ -3 & 0 & -1\end{array}\right]$
Conclusion: $C=\left[\begin{array}{lll}-3 & 4 & -1 \\ -3 & 0 & -1\end{array}\right]$
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