If the matrix $A=\begin{bmatrix} 0 & 2 \\ K & -1 \end{bmatrix}$ satisfies $A(A^{3}+3I)=2I$,then the value of $K$ is:

If $A=\begin{bmatrix} \sqrt{2} & 1 \\ -1 & \sqrt{2} \end{bmatrix}$,$B=\begin{bmatrix} 1 & 0 \\ 1 & 1 \end{bmatrix}$,$C=ABA^T$ and $X=A^T C^2 A$,then $\operatorname{det}(X)$ is equal to:

$A$ and $B$ are $3 \times 3$ matrices such that $AB + A + B = 0$,then:

If $a_{r}=(\cos 2 r \pi+i \sin 2 r \pi)^{1 / 9}$, then the value of $\left|\begin{array}{lll}a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9}\end{array}\right|$ is equal to

Let $p$ be a non-singular matrix such that $I + p + p^2 + .... + p^n = O$ (where $O$ denotes the null matrix and $I$ denotes the identity matrix),then $p^{-1} = $

If $\left[\begin{array}{ccc}0 & 2 & a \\ b & 0 & 4 \\ -3 & c & 0\end{array}\right]$ is a skew-symmetric matrix, then $\left[\begin{array}{cc}a & b \\ b & a\end{array}\right]\left[\begin{array}{cc}b & c \\ c & b\end{array}\right]=$