If $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$,$10 B = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{bmatrix}$ and $B = A^{-1}$,then the value of $\alpha$ is:

The inverse of the matrix $A = \begin{bmatrix} 3 & -2 \\ 1 & 4 \end{bmatrix}$ is:

If $A$ and $B$ are non-singular square matrices of the same order,then $adj(AB)$ is equal to:

Let $A$ be a $n \times n$ matrix such that $A$ is an upper-triangular matrix. Then $adj(A) =$

Which of the following matrices are invertible?
$A = \begin{bmatrix} 2 & 3 \\ 10 & 15 \end{bmatrix}, B = \begin{bmatrix} 1 & 2 & 3 \\ 2 & -1 & 3 \\ 1 & 2 & 3 \end{bmatrix}, C = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8 \end{bmatrix}, D = \begin{bmatrix} 2 & 4 & 2 \\ 1 & 1 & 0 \\ 1 & 4 & 5 \end{bmatrix}$

Let $A$ be a $n \times n$ matrix such that $|A|=2$. If the determinant of the matrix $\operatorname{Adj}(2 \cdot \operatorname{Adj}(2A^{-1}))$ is $2^{84}$,then $n$ is equal to: