Let $A = \begin{bmatrix} 1 & 2 \\ -1 & 4 \end{bmatrix}$. If $A^{-1} = \alpha I + \beta A$,where $\alpha, \beta \in \mathbb{R}$ and $I$ is a $2 \times 2$ identity matrix,then $4(\alpha - \beta)$ is equal to:

If $A=\left[\begin{array}{ll}2 & 3 \\ 1 & 2\end{array}\right]$ and $B=\left[\begin{array}{cc}2 & -3 \\ -1 & 2\end{array}\right]$,then $\left(B^{-1} A^{-1}\right)^{-1}=$

If $A = \begin{bmatrix} a & b & c \\ d & e & f \\ l & m & n \end{bmatrix}$ is a matrix such that $|A| > 0$ and $\text{Adj}(A) = \begin{bmatrix} 0 & 4 & -6 \\ 10 & 8 & 0 \\ 2 & 4 & -4 \end{bmatrix}$,then $\frac{cd}{fb} + \frac{\ln}{em} = $

Let $A = \begin{bmatrix} 1 & -2 & 1 \\ -2 & 3 & 1 \\ 1 & 1 & 5 \end{bmatrix}$. Verify that $[adj A]^{-1} = adj(A^{-1})$.

If $A = \begin{bmatrix} 2 & -1 \\ -1 & 2 \end{bmatrix}$ such that $A^2 - 4A + 3I = 0$,where $I$ is a unit matrix of order $2$,then $A^{-1}$ is

The element in the third row and the second column in the inverse matrix of a matrix $A = \begin{bmatrix} 1 & 3 & 3 \\ 3 & 1 & 3 \\ 3 & 3 & 4 \end{bmatrix}$ is