Suppose $A$ is any $3 \times 3$ non-singular matrix and $(A-3 I)(A-5 I)=O$,where $I=I_3$ and $O=O_3$. Here $O_3$ represents the zero matrix of order $3$ and $I_3$ is the identity matrix of order $3$. If $\alpha A+\beta A^{-1}=4 I$,then $\alpha+\beta$ is equal to:

If $A = \begin{bmatrix} 1 & \tan \frac{\alpha}{2} \\ -\tan \frac{\alpha}{2} & 1 \end{bmatrix}$ and $AB = I$,then $B$ is equal to

Find the inverse of the matrix,if it exists: $\left[\begin{array}{ll}1 & 3 \\ 2 & 7\end{array}\right]$

The element in the third row and first column of the inverse of the matrix $\left[\begin{array}{ccc}1 & -3 & 2 \\ -3 & 3 & -1 \\ 2 & -1 & 0\end{array}\right]$ is

Let $A = \begin{bmatrix} 2k - 1 & 1 & 1 \\ 0 & 2k - 1 & 1 \\ 0 & 0 & 2k - 1 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & 2k - 1 & 1 \\ 1 - 2k & 0 & k \\ -1 & -k & 0 \end{bmatrix}$ where $k$ is a real number. If $\det(\text{adj } A) + \det(\text{adj } B) = 11^6$, then the value of $k - 5$ is equal to...

If $A = \begin{bmatrix} k & 5 & 2 \\ 2 & -k & 5 \\ 5 & 2 & -k \end{bmatrix}$ and $\det A = 190$, then $\operatorname{Adj} A = $