Find the adjoint of the matrix: $\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$

If $A = \begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix}$,then $A^{-1}$ is

Let $A = \begin{bmatrix} -1 & 1 & -1 \\ 1 & 0 & 1 \\ 0 & 0 & 1 \end{bmatrix}$ satisfy $A^2 + \alpha(adj(adj(A))) + \beta(adj(A)(adj(adj(A)))) = \begin{bmatrix} 2 & -2 & 2 \\ -2 & 0 & -1 \\ 0 & 0 & -1 \end{bmatrix}$ for some $\alpha, \beta \in R$. Then $(\alpha - \beta)^2$ is equal to . . . . . . .

If $\omega$ is a complex cube root of unity and $A=\begin{bmatrix} \omega & 0 \\ 0 & \omega \end{bmatrix}$,then $A^{-1}=$

Let the determinant of a square matrix $A$ of order $m$ be $m-n$,where $m$ and $n$ satisfy $4m + n = 22$ and $17m + 4n = 93$. If $\operatorname{det}(n \operatorname{adj}(\operatorname{adj}(mA))) = 3^a 5^b 6^c$,then $a + b + c$ is equal to:

If $A$ is a square matrix satisfying the equation $A^2 - 5A + 7I = 0$,where $I$ is the identity matrix and $0$ is the null matrix of the same order,then $A^{-1} = $