If $A, B, C$ are three square matrices such that $AB = AC$ implies $B = C$,then the matrix $A$ is always a/an

If $A=\begin{bmatrix} 1 & 0 & -2 \\ -2 & -1 & 2 \\ 3 & 4 & 1 \end{bmatrix}$,then $A^{-1}=$

Let $A$ be a $3 \times 3$ matrix and $\det(A)=2$. If $n = \det(\underbrace{\operatorname{adj}(\operatorname{adj}(\ldots(\operatorname{adj} A)))}_{2024 \text{ times}})$,then the remainder when $n$ is divided by $9$ is equal to

Let $A$ be a $3 \times 3$ matrix such that $\operatorname{adj} A = \begin{bmatrix} 2 & -1 & 1 \\ -1 & 0 & 2 \\ 1 & -2 & -1 \end{bmatrix}$ and $B = \operatorname{adj}(\operatorname{adj} A)$. If $|A| = \lambda$ and $|(B^{-1})^T| = \mu$,then the ordered pair $(|\lambda|, \mu)$ is equal to

If $F(\alpha ) = \begin{bmatrix} \cos \alpha & - \sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$ and $G(\beta ) = \begin{bmatrix} \cos \beta & 0 & \sin \beta \\ 0 & 1 & 0 \\ - \sin \beta & 0 & \cos \beta \end{bmatrix}$,then $[F(\alpha ) G(\beta )]^{-1} = $

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