If $A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 3 \\ 1 & 0 & 1 \end{bmatrix}$,then $|\operatorname{adj} A| = $ . . . . . . .

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

If matrix $A = \begin{bmatrix} 3 & 2 & 4 \\ 1 & 2 & -1 \\ 0 & 1 & 1 \end{bmatrix}$ and $A^{-1} = \frac{1}{K} \text{adj}(A)$,then $K$ is:

If $X$ is a square matrix of order $3 \times 3$ and $\lambda$ is a scalar,then $adj(\lambda X)$ is equal to

If $A = \begin{bmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{bmatrix}$,then the matrix $A^{-50}$ when $\theta = \frac{\pi}{12}$ is equal to:

If $A = \begin{bmatrix} 1 & -1 & 1 \\ 2 & 1 & -3 \\ 1 & 1 & 1 \end{bmatrix}$,$10 B = \begin{bmatrix} 4 & 2 & 2 \\ -5 & 0 & \alpha \\ 1 & -2 & 3 \end{bmatrix}$ and $B = A^{-1}$,then the value of $\alpha$ is: