If $A$ is a skew-symmetric matrix and $n$ is a positive integer,then $A^n$ is

If the elements of matrix $A$ are the reciprocals of elements of matrix $\left[\begin{array}{ccc}1 & \omega & \omega^{2} \\ \omega & \omega^{2} & 1 \\ \omega^{2} & 1 & \omega\end{array}\right]$,where $\omega$ is a complex cube root of unity,then:

Let $M=\begin{bmatrix} \sin^4 \theta & -1-\sin^2 \theta \\ 1+\cos^2 \theta & \cos^4 \theta \end{bmatrix} = \alpha I + \beta M^{-1}$,where $\alpha = \alpha(\theta)$ and $\beta = \beta(\theta)$ are real numbers,and $I$ is the $2 \times 2$ identity matrix. If $\alpha^*$ is the minimum of the set $\{\alpha(\theta) : \theta \in [0, 2\pi)\}$ and $\beta^*$ is the minimum of the set $\{\beta(\theta) : \theta \in [0, 2\pi)\}$,then the value of $\alpha^* + \beta^*$ is

$A$ and $B$ are two square matrices such that $A^2B = BA$. If $(AB)^{10} = A^K B^{10}$,then $k$ is:

The value of $\theta$ lying between $-\frac{\pi}{4}$ and $\frac{\pi}{2}$ and $0 \le A \le \frac{\pi}{2}$ satisfying the equation $\begin{vmatrix} 1 + \sin^2 A & \cos^2 A & 2 \sin 4\theta \\ \sin^2 A & 1 + \cos^2 A & 2 \sin 4\theta \\ \sin^2 A & \cos^2 A & 1 + 2 \sin 4\theta \end{vmatrix} = 0$ are:

The number of solutions of the matrix equation $X^2 = \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix}$ is