If $B = \begin{bmatrix} 5 & 2\alpha & 1 \\ 0 & 2 & 1 \\ \alpha & 3 & -1 \end{bmatrix}$ is the inverse of a $3 \times 3$ matrix $A$,then the sum of all values of $\alpha$ for which $\det(A) + 1 = 0$ is:

The inverse of the matrix $\left[\begin{array}{ccc}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$ is

If $A=\left[\begin{array}{ll}2 & -2 \\ 2 & -3\end{array}\right]$ and $B=\left[\begin{array}{cc}0 & -1 \\ 1 & 0\end{array}\right]$,then $(B^{-1} A^{-1})^{-1} = ?$

If $A = \begin{bmatrix} a+ib & c+id \\ -c+id & a-ib \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} a+ib & -c-id \\ -c+id & a-ib \end{bmatrix}$,find $(a^2+b^2+c^2+d^2)$.

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} = $

If $A$ is an invertible matrix of order $n$,then the determinant of $\operatorname{adj} A$ is equal to :