If M= bmatrix 2 & 0 & 0 0 & 2 & 0 0 & 0 & 2 bmatrix , then what is...

Given $M = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{bmatrix}$

The determinant of the diagonal matrix $M$ is $|M| = 2 \times 2 \times 2 = 8$.

For a square matrix of order $n$, the determinant of its adjoint is given by $|\text{adj}M| = |M|^{n-1}$.

Since the order of $M$ is $n = 3$, we have $|\text{adj}M| = |M|^{3-1} = |M|^2$.

The required value is $|M| \, |\text{adj}M| = |M| \times |M|^2 = |M|^3$.

Substituting $|M| = 8$, we get $8^3 = 512$.

Answer: $512$