Power 4 Determinant!

For a square matrix 'A' of order nxn or simply 'n' and a constant 'k', $|kA|=(k^{n})|A|$ & $|adj(A)|={|A|}^{n-1}$

$|\frac{1}{4}A|=4$ Therefore, $|\frac{1}{4}A|=(\frac{1}{4})^{n}|A|=4$ $|A|=1024=2^{10}=4^5$ $\Rightarrow \frac{1}{4^n}.4^5=4 \Rightarrow n=4$

Therefore, $|adj(A)|={|A|}^{n-1}=(4^5)^{4-1}=4^{3*5}=\boxed{4^{15}}$

$|\frac{1}{4}A|=4$

$(\frac{1}{4})^{n}$ $|A|=4$

$4^{-n}$ $\times$ $4^5=4$

$-n+5=1$

$n=4$

So , $|adj~A|=|A|^{n-1}=(4^{5})^{3}=4^{15}$