JEE Main · 2021 · 17 Mar 2021 · Shift 1 · Matrices

Here $\det A=2(-1)-3(0)=-2$, so $\det \left(A^4\right)=(-2{)}^4=16$.
For a $2\times 2$ matrix $\mathrm{Adj}(2A)=2\mathrm{Adj}(A)$, and $\mathrm{Adj}(A)=\left(\begin{array}{cc}-1& -3\\ 0& 2\end{array}\right)$, so $\mathrm{Adj}(2A)=\left(\begin{array}{cc}-2& -6\\ 0& 4\end{array}\right)$.
Both $A$ and $\mathrm{Adj}(2A)$ are upper triangular, and a power of an upper triangular matrix is upper triangular with the corresponding powers of the diagonal entries:

The two leading entries are equal and both lower-left entries vanish, so the first column of $A^{10}-{\left(\mathrm{Adj}(2A)\right)}^{10}$ is zero and its determinant is $0$. The required value is $16+0=16$.