For $a, b > 0$, if $P =\begin{bmatrix}0&-a\\2a&b\end{bmatrix}$ and $Q=\begin{bmatrix}b&a\\-b&0\end{bmatrix}$ are two matrices such that $PQ =\begin{bmatrix}2&0\\3&8\end{bmatrix}$, then the value of $(a+b)^{ab}$ is:

9

The correct answer is Option (2) → 9

$PQ =\begin{bmatrix}0&-a\\2a&b\end{bmatrix}\begin{bmatrix}b&a\\-b&0\end{bmatrix}$

$=\begin{bmatrix}ab&0\\2ab-b^2&2a^2\end{bmatrix}=\begin{bmatrix}2&0\\3&8\end{bmatrix}$

$⇒ab=2$

$⇒2a^2=8$

$⇒a=2$

$⇒b=1$

$∴(a+b)^{ab}=(3)^2=9$