If $P=\begin{bmatrix} -2 & 2 & 0\\3 & 1 & 4\end{bmatrix}$ and $Q=\begin{bmatrix} 2 & 0 & 2\\7 & 1 & 6\end{bmatrix}$. If $5Q-3P+2R=0$ then the matrix R is _________.

$\begin{bmatrix} -8 & 3 & 5\\-13 & -1 & -9\end{bmatrix}$

The correct answer is Option (3) → $\begin{bmatrix} -8 & 3 & 5\\-13 & -1 & -9\end{bmatrix}$

$P=\begin{bmatrix} -2 & 2 & 0\\3 & 1 & 4\end{bmatrix}$ and $Q=\begin{bmatrix} 2 & 0 & 2\\7 & 1 & 6\end{bmatrix}$.

$5Q-3P+2R=0$

$2R=-5Q+3P⇒R=\frac{-5Q+3P}{2}$

$⇒-5Q=\begin{bmatrix} -10 & 0 & -10\\-35 & -5 & -30\end{bmatrix}$

$⇒3P=\begin{bmatrix} -6 & 6 & 0\\9 & 3 & 12\end{bmatrix}$

$⇒-5Q+3P=\begin{bmatrix} -10 & 0 & -10\\-35 & -5 & -30\end{bmatrix}+\begin{bmatrix} -6 & 6 & 0\\9 & 3 & 12\end{bmatrix}=\begin{bmatrix} -16 & 6 & -10\\-26 & -2 & -18\end{bmatrix}$

$⇒R=\frac{1}{2}\begin{bmatrix} -16 & 6 & -10\\-26 & -2 & -18\end{bmatrix}=\begin{bmatrix} -8 & 3 & 5\\-13 & -1 & -9\end{bmatrix}$