If $A = \begin{bmatrix} 2 & 3 \\ 4 & 6 \end{bmatrix}$,then ${A^{-1}} = $

Inverse of the matrix $\begin{bmatrix} \cos 2\theta & -\sin 2\theta \\ \sin 2\theta & \cos 2\theta \end{bmatrix}$ is

If $A$ is a square matrix of order $3$,then $|\operatorname{Adj}(\operatorname{Adj} A^2)|=$

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 0 & 2 & -3 \\ 2 & 1 & 3 \end{bmatrix}$ and $B = \operatorname{adj} A$,$C = 5A$,then $\frac{|\operatorname{adj} B|}{|C|} = $

Let $A$ be a $3 \times 3$ matrix such that $|\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A ))|=81$. If $S =\{ n \in \mathbb{Z} :(|\operatorname{adj}(\operatorname{adj} A)|)^{\frac{(n-1)^2}{2}}=|A|^{(3n^2-5n-4)}\}$,then $\sum_{n \in S}|A^{(n^2+n)}|$ is equal to

If $A$ and $B$ are non-singular matrices and $\operatorname{det}(AB)=(\operatorname{det} A)(\operatorname{det} B)$,then $((\operatorname{det} A)(\operatorname{det} B)) B^{-1} A^{-1} =$