The set of all $2 \times 2$ matrices over the real numbers is not a group under matrix multiplication because

If $A$ and $B$ are square matrices of the same order and $AB = 3I$, then $A^{-1}$ is equal to

Let $X=\begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{bmatrix}$,$Y=\alpha I+\beta X+\gamma X^{2}$ and $Z=\alpha^{2} I-\alpha \beta X+\left(\beta^{2}-\alpha \gamma\right) X^{2}$,where $\alpha, \beta, \gamma \in \mathbb{R}$. If $Y^{-1}=\begin{bmatrix} \frac{1}{5} & \frac{-2}{5} & \frac{1}{5} \\ 0 & \frac{1}{5} & \frac{-2}{5} \\ 0 & 0 & \frac{1}{5} \end{bmatrix}$,then $(\alpha-\beta+\gamma)^{2}$ is equal to

Let $P=\begin{bmatrix} 3 & -1 & -2 \\ 2 & 0 & \alpha \\ 3 & -5 & 0 \end{bmatrix}$,where $\alpha \in \mathbb{R}$. Suppose $Q=[q_{ij}]$ is a matrix such that $PQ=kI$,where $k \in \mathbb{R}, k \neq 0$ and $I$ is the identity matrix of order $3$. If $q_{23}=-\frac{k}{8}$ and $\det(Q)=\frac{k^2}{2}$,then:

If a matrix $A$ is such that $4A^3 + 2A^2 + 7A + I = O$,then $A^{-1}$ equals

Find the inverse,by elementary row operations (if possible),of the following matrix: $\left[\begin{array}{cc}1 & -3 \\ -2 & 6\end{array}\right]$