The total number of matrices $A = \begin{bmatrix} 0 & 2x & 2x \\ 2y & y & -y \\ 1 & -1 & 1 \end{bmatrix}$ where $x, y \in \mathbb{R}$ and $x \neq y$,for which $A^T A = 3I_3$ is:

Let $X = \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}$. Let $Y$ be a $2 \times 2$ real matrix satisfying the condition $XY = YX$. Then the smallest possible value of $\det(Y)$ is

Let $A = \begin{bmatrix} 1 & 2 & 3 \\ 2 & 2 & -1 \\ 3 & 0 & k \end{bmatrix}$ and $f(x) = x^3 - 2x^2 - \alpha x + \beta = 0$. If $A$ satisfies $f(A) = 0$,then:

If the determinant $\Delta = \begin{vmatrix} a & b & a\alpha + b \\ b & c & b\alpha + c \\ a\alpha + b & b\alpha + c & 0 \end{vmatrix} = 0$,then:

The least positive integer $n$ such that $\left(\begin{array}{cc}\cos \frac{\pi}{4} & \sin \frac{\pi}{4} \\ -\sin \frac{\pi}{4} & \cos \frac{\pi}{4}\end{array}\right)^{n}$ is an identity matrix of order $2$ is

If $\alpha = \cos \frac{\pi}{3} + i \sin \frac{\pi}{3}$,then the value of the determinant $\left| \begin{array}{ccc} 1 & \alpha & \alpha^2 \\ \alpha^2 & 1 & \alpha \\ \alpha & \alpha^2 & 1 \end{array} \right|$ is