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Question
If A = `[[1 -3 2],[2 0 2]]`and `B = [[2 -1 -1],[1 0 -1]]` find the matrix C such that A + B + C is
, find the matrix C such that A + B + C is zero matrix.
Solution
\[Given: \hspace{0.167em} A + B + C = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}1 & - 3 & 2 \\ 2 & 0 & 2\end{bmatrix} + \begin{bmatrix}2 & - 1 & - 1 \\ 1 & 0 & - 1\end{bmatrix} + C = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}1 + 2 & - 3 - 1 & 2 - 1 \\ 2 + 1 & 0 + 0 & 2 - 1\end{bmatrix} + C = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]
\[ \Rightarrow \begin{bmatrix}3 & - 4 & 1 \\ 3 & 0 & 1\end{bmatrix} + C = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix}\]
\[ \Rightarrow C = \begin{bmatrix}0 & 0 & 0 \\ 0 & 0 & 0\end{bmatrix} - \begin{bmatrix}3 & - 4 & 1 \\ 3 & 0 & 1\end{bmatrix}\]
\[ \Rightarrow C = \begin{bmatrix}0 - 3 & 0 + 4 & 0 - 1 \\ 0 - 3 & 0 - 0 & 0 - 1\end{bmatrix}\]
\[ \Rightarrow C = \begin{bmatrix}- 3 & 4 & - 1 \\ - 3 & 0 & - 1\end{bmatrix}\]
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