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

The element in the $1^{st}$ row and $2^{nd}$ column of the inverse of the matrix $A = \begin{bmatrix} 1 & 3 & -2 \\ -3 & 0 & -5 \\ 2 & 5 & 0 \end{bmatrix}$ is ...

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:

Find the inverse of each of the matrices (if it exists). $\left[\begin{array}{ccc}1 & 0 & 0 \\ 3 & 3 & 0 \\ 5 & 2 & -1\end{array}\right]$

If $|A| = 2$,where $A$ is a square matrix of order $4$,then the value of $|Adj(Adj(2A))|$ is (where $Adj(A)$ denotes the adjoint of matrix $A$):

If for a matrix $A$, $|A|=6$ and $\text{adj } A = \begin{bmatrix} 1 & -2 & 4 \\ 4 & 1 & 1 \\ -1 & k & 0 \end{bmatrix}$, then $k$ is equal to