if

Question:

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

Solution:

Given : $A+B+C=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

$\Rightarrow\left[\begin{array}{ccc}1 & -3 & 2 \\ 2 & 0 & 2\end{array}\right]+\left[\begin{array}{ccc}2 & -1 & -1 \\ 1 & 0 & -1\end{array}\right]+C=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

$\Rightarrow\left[\begin{array}{ccc}1+2 & -3-1 & 2-1 \\ 2+1 & 0+0 & 2-1\end{array}\right]+C=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

$\Rightarrow\left[\begin{array}{ccc}3 & -4 & 1 \\ 3 & 0 & 1\end{array}\right]+C=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]$

$\Rightarrow C=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]-\left[\begin{array}{ccc}3 & -4 & 1 \\ 3 & 0 & 1\end{array}\right]$

$\Rightarrow C=\left[\begin{array}{lll}0-3 & 0+4 & 0-1 \\ 0-3 & 0-0 & 0-1\end{array}\right]$

$\Rightarrow C=\left[\begin{array}{lll}-3 & 4 & -1 \\ -3 & 0 & -1\end{array}\right]$

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