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

If $A=\left[\begin{array}{ccc}2 & -1 & 1 \\ -1 & 2 & -1 \\ 1 & -1 & 2\end{array}\right]$,verify that $A^{3}-6 A^{2}+9 A-4 I=0$ and hence find $A^{-1}$.

The inverse of the matrix $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ is

Let $A$ be a $4 \times 4$ matrix and $P$ be its adjoint matrix. If $|P|=\left|\frac{A}{2}\right|$,then $\left|A^{-1}\right|=$

If $A = \begin{bmatrix} 2 & 0 & -1 \\ 5 & 1 & 0 \\ 0 & 1 & 3 \end{bmatrix}$ and $A^{-1} = \begin{bmatrix} 3 & -1 & 1 \\ \alpha & 6 & -5 \\ \beta & -2 & 2 \end{bmatrix}$,then the values of $\alpha$ and $\beta$ are respectively:

If $A = \begin{bmatrix} i & 0 \\ 0 & i/2 \end{bmatrix}$ where $i = \sqrt{-1}$,then $A^{-1} = $