If $A^2-A+I=0$, then the inverse of the matrix $A$ is

Find the inverse of the matrix,if it exists: $\left[\begin{array}{ll}3 & 1 \\ 5 & 2\end{array}\right]$

If $A(\operatorname{adj} A)=5 I$ where $I$ is the identity matrix of order $3$,then $|\operatorname{adj} A|$ is equal to

Let $A = \begin{bmatrix} -\cot \theta & \operatorname{cosec} \theta \\ \operatorname{cosec} \theta & -\cot \theta \end{bmatrix}$. If $A^{-1} = A$ at $\theta = \theta_1$ and $A^{-1} + A = O$ at $\theta = \theta_2$,then which one of the following is true?

If $A = \begin{bmatrix} \sec \theta & -\tan \theta \\ -\tan \theta & \sec \theta \end{bmatrix}$, $\theta \in (0, \frac{\pi}{2})$ such that $A + \text{adj } A = 4I$, then $\theta =$

If $A=\begin{bmatrix} 2a & -3b \\ 3 & 2 \end{bmatrix}$ and $A \cdot \operatorname{adj} A = A A^{T}$,then $2a + 3b$ is