Question

Let matrix $A=\begin{bmatrix} x & y & -z \\ 1 & 2 & \, \, 3 \\ 1 & 1 & \, \, 2 \end{bmatrix}$ , 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

$\operatorname{det} \cdot(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A ))))=| A |^{(3-1)^{4}}$
$=\left|A\right|^{16}=4^{8}. 5^{16}$
$\Rightarrow \left|A\right|=\pm10$
$\left|A\right|=\begin{vmatrix} x & y & -z \\ 1 & 2 & 3 \\ 1 & 1 & 2 \end{vmatrix}=x+y+z=\pm10$
$\because x,y,z\in N\Rightarrow x+y+z=-10$ (not possible)
Hence, $x+y+z=10$
The number of such matrices $= ^{9}C_{2}$
$=36$