Let $A = \begin{bmatrix} 4 & -2 \\ \alpha & \beta \end{bmatrix}$. If $A^2 + \gamma A + 18I = O$,then $\operatorname{det}(A)$ is equal to

Let $A$ and $B$ be two square matrices of order $3$ such that $|A|=3$ and $|B|=2$. Then $\left|A^{T} A(\operatorname{adj}(2A))^{-1}(\operatorname{adj}(4B))(\operatorname{adj}(AB))^{-1} AA^{T}\right|$ is equal to:

If $A = \begin{bmatrix} 3 & 7 \\ 1 & 2 \end{bmatrix}$,then $|A^{2011} - 5A^{2010}|$ is equal to

Match the items of List-$I$ with the items of List-$II$ and choose the correct option:

List-$I$	List-$II$
$(A)$ If $A$ is a non-singular matrix of order $3$ and $|A|=a$,then $|\text{adj}(A)|=$	$(I)$ null matrix
$(B)$ $A$ is a non-singular matrix of order $3$ and $B$ is any matrix of order $3$ such that $AB=O$,then $B$ is	$(II)$ $a^2$
$(C)$ $\begin{vmatrix} 1 & x & x^2 \\ \cos(a-b)y & \cos ay & \cos(a+b)y \\ \sin(a-b)y & \sin ay & \sin(a+b)y \end{vmatrix}$ does not depend on	$(III)$ $b$
$(D)$ $A$ is a square matrix of order $3$ and $B=A-A^T$,then $B$ is	$(IV)$ $a$
	$(V)$ $0$

If $A$ and $B$ are $3 \times 3$ order matrices and $|A|=5$,$|B|=3$,then $|3AB|=$ . . . . . . .

If $A$ and $B$ are square matrices of order $3$ such that $(A + B)(A - B) = A^2 - B^2$,then $(ABA^{-1})^2$ is equal to