If $A=\begin{bmatrix}1 & 5\\7 & 12\end{bmatrix}$ and $B=\begin{bmatrix}9 & 1\\7 & 8 \end{bmatrix}, $ then matrix C for which $3A+ 5B + 2C$ is a null matrix is :

$\begin{bmatrix}-24 & -10\\-28 & -38\end{bmatrix}$

The correct answer is Option (3) → $\begin{bmatrix}-24 & -10\\-28 & -38\end{bmatrix}$

$3A+5B=3\begin{bmatrix}1 & 5\\7 & 12\end{bmatrix}+5\begin{bmatrix}9 & 1\\7 & 8 \end{bmatrix}$

$=\begin{bmatrix}48 & 20\\56 & 76\end{bmatrix}$

$⇒C=-\frac{1}{2}[3A+5B]$

$⇒C=-\frac{1}{2}\begin{bmatrix}48 & 20\\56 & 76\end{bmatrix}$

$=\begin{bmatrix}-24 & -10\\-28 & -38\end{bmatrix}$