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Question:

If $A=\left[\begin{array}{ll}9 & 1 \\ 7 & 8\end{array}\right], B=\left[\begin{array}{cc}1 & 5 \\ 7 & 12\end{array}\right]$, find matrix $C$ such that $5 A+3 B+2 C$ is a null matrix.

Solution:

Given : $5 A+3 B+2 C=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

$\Rightarrow 5\left[\begin{array}{ll}9 & 1 \\ 7 & 8\end{array}\right]+3\left[\begin{array}{cc}1 & 5 \\ 7 & 12\end{array}\right]+2 C=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}45 & 5 \\ 35 & 40\end{array}\right]+\left[\begin{array}{cc}3 & 15 \\ 21 & 36\end{array}\right]+2 C=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

$\Rightarrow\left[\begin{array}{cc}45+3 & 5+15 \\ 35+21 & 40+36\end{array}\right]+2 C=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

$\Rightarrow\left[\begin{array}{ll}48 & 20 \\ 56 & 76\end{array}\right]+2 C=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]$

$\Rightarrow 2 C=\left[\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right]-\left[\begin{array}{ll}48 & 20 \\ 56 & 76\end{array}\right]$

$\Rightarrow C=\frac{1}{2}\left[\begin{array}{ll}-48 & -20 \\ -56 & -76\end{array}\right]$

$\Rightarrow C=\left[\begin{array}{ll}-24 & -10 \\ -28 & -38\end{array}\right]$

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