Find $P^{-1},$ if it exists,given $P=\left[\begin{array}{cc}10 & -2 \\ -5 & 1\end{array}\right].$

Let $\alpha, \beta, \gamma$ be real numbers. If $A=\begin{bmatrix} 7 & 3 & \alpha \\ \beta & 1 & -11 \\ -5 & \gamma & 19 \end{bmatrix}$ is a $3 \times 3$ matrix satisfying $A\begin{bmatrix} 5 \\ -13 \\ 11 \end{bmatrix}=\begin{bmatrix} -290 \\ -119 \\ 210 \end{bmatrix}$, then $(\operatorname{adj} A)^{-1}+\operatorname{adj} A^{-1}=$

If $A=\begin{bmatrix} 2 & -3 \\ 5 & -7 \end{bmatrix}$,then $A-A^{-1}=$

If $A = \begin{bmatrix} 5a & -b \\ 3 & 2 \end{bmatrix}$ and $A (adj A) = AA^T$, then $5a + b =$

If $A = \begin{bmatrix} 2 & -1 \\ -1 & 3 \end{bmatrix}$,then the inverse of $(2A^2 + 5A)$ is

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