If $|\operatorname{Adj} A|=x$ and $|\operatorname{Adj} B|=y$,then $\left|(\operatorname{Adj}(AB))^{-1}\right|=$

For any $2 \times 2$ matrix $A$,if $A(\text{adj } A) = \begin{bmatrix} 10 & 0 \\ 0 & 10 \end{bmatrix}$,then $|A|$ is equal to:

If $A$ is a square matrix of order $3$,then consider the following statements.
$I$. If $|A|=0$,then $|\operatorname{Adj} A|=0$
$II$. If $|A| \neq 0$,then $|A^{-1}|=|A|^{-1}$
Which of the above statements is/are true?

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 0 \end{bmatrix}$,$B = \operatorname{adj} A$ and $C = 5A$,then find the value of $\frac{|\operatorname{adj} B|}{|C|}$.

Let $A = \begin{bmatrix} x + \lambda & x & x \\ x & x + \lambda & x \\ x & x & x + \lambda \end{bmatrix}$,then $A^{-1}$ exists if

If matrix $A = \begin{bmatrix} 3 & 2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{K} \text{adj}(A)$,then $K$ is: