Question

Let matrix $A=\left[\begin{array}{ccc}x& y& -z\\ 1& 2& 3\\ 1& 1& 2\end{array}\right]$ , where $\text{x},\text{y},\text{z}\in N$ . If $\left|adj\left|adj\left|adj\left|adjA\right|\right|\right|\right|=4^8\cdot 5^{16}$ , then the number of such matrices $A$ is equal to (where, $\left|M\right|$ represents determinant of a matrix $M$ )

Answer 1

Answer: 36

$\det \cdot (\mathrm{adj}(\mathrm{adj}(\mathrm{adj}(\mathrm{adj}A))))=|A{|}^{(3-1{)}^4}$
$={\left|A\right|}^{16}=4^8.5^{16}$
$\Rightarrow \left|A\right|=\pm 10$
$\left|A\right|=\left|\begin{array}{ccc}x& y& -z\\ 1& 2& 3\\ 1& 1& 2\end{array}\right|=x+y+z=\pm 10$
$\because x,y,z\in N\Rightarrow x+y+z=-10$ (not possible)
Hence, $x+y+z=10$
The number of such matrices ${=}^9{C}_2$
$=36$