Let the matrices $A$ and $B$ be defined as $A=$
$\begin{bmatrix}
3 & 2\\
2 & 1\\
\end{bmatrix}$ and $B=$
$\begin{bmatrix}
3 & 1\\
7 & 3\\
\end{bmatrix}$ then the value of $|2A^{9}B^{-1}|$ is:
$(A)$ $2$ $(B)$ $1$ $(C)$ $-1$ $(D)$ $-2$.
In the above question do we have to solve each and every term manually as power of A is $9$ or is there some other method , shortcut or a trick? Thank you.
Provided we are dealing with square matrices, then $\det(A B) = \det(A) \det(B)$. Thus, $$\det(2 I A^9 B^{-1}) = \det(2 I) \det(A)^9 \det(B)^{-1}.$$ So no, you don't need to compute $A^9$ or even $B^{-1}$ for that matter.
Further hint: Be careful finding $\det(2I)$. It is probably not what you think it is.