Question

If A,B,C are three square matrices of third order such that $A=\left[\begin{array}{ccc}x& 0& 2\\ 0& y& 0\\ 0& 0& z\end{array}\right]$ & $|B|=2^2{3}^2,|C|=2,x,y,z\in I^{+}$ & $|adj\left(adjABC\right)|=2^{16}{3}^8{7}^4$ then number of distinct possible matrices A is

Answer 1

$|A{|}^4|B{|}^4|C{|}^4=2^{16}{3}^8{7}^4$
$|A{|}^4=2^4{7}^4$
|A| =xyz
$xyz=2\times 7$
$(x,y,z)\to \underset{\underset{6ways}{\downarrow }}{(2,1,7)},\underset{\underset{3ways}{\downarrow }}{\underset{}{(14,1,1)}}$
Total $\to 9$ ways