If $A = \begin{bmatrix} \cos^2 \alpha & \sin \alpha \cos \alpha \\ \sin \alpha \cos \alpha & \sin^2 \alpha \end{bmatrix}$ and $B = \begin{bmatrix} \cos^2 \beta & \sin \beta \cos \beta \\ \sin \beta \cos \beta & \sin^2 \beta \end{bmatrix}$ are such that $AB$ is a null matrix,then which of the following must be an odd integral multiple of $\frac{\pi}{2}$?

Let $A$ and $B$ be two $3 \times 3$ matrices such that $AB = I$ and $|A| = \frac{1}{8}$. Then $|\operatorname{adj}(B \operatorname{adj}(2A))|$ is equal to:

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A)$ the sum of diagonal entries of $A$. Assume that $A^2 = I$.
Statement-$1$: If $A \neq I$ and $A \neq -I$,then $\det(A) = -1$.
Statement-$2$: If $A \neq I$ and $A \neq -I$,then $tr(A) \neq 0$.

Suppose the vectors $x_{1}, x_{2}$ and $x_{3}$ are the solutions of the system of linear equations $Ax = b$ when the vector $b$ on the right side is equal to $b_{1}, b_{2}$ and $b_{3}$ respectively. If $x_{1} = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, x_{2} = \begin{bmatrix} 0 \\ 2 \\ 1 \end{bmatrix}, x_{3} = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, b_{1} = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, b_{2} = \begin{bmatrix} 0 \\ 2 \\ 0 \end{bmatrix}$ and $b_{3} = \begin{bmatrix} 0 \\ 0 \\ 2 \end{bmatrix}$,then the determinant of $A$ is equal to

Let $M$ be a $2 \times 2$ symmetric matrix with integer entries. Then $M$ is invertible if:

The equation $\left| \begin{array}{ccc} (1+x)^2 & (1-x)^2 & -(2+x^2) \\ 2x+1 & 3x & 1-5x \\ x+1 & 2x & 2-3x \end{array} \right| + \left| \begin{array}{ccc} (1+x)^2 & 2x+1 & x+1 \\ (1-x)^2 & 3x & 2x \\ 1-2x & 3x-2 & 2x-3 \end{array} \right| = 0$