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 $S =\{ M = [a_{ij}], a_{ij} \in \{0,1,2\}, 1 \leq i, j \leq 2\}$ be a sample space and $A = \{M \in S : M \text{ is invertible}\}$ be an event. Then $P(A)$ is equal to

If $A$ and $B$ are two non-zero $n \times n$ matrices such that $A^2 + B = A^2 B$,then:

Let $A$ and $B$ be $3 \times 3$ real matrices such that $A$ is a symmetric matrix and $B$ is a skew-symmetric matrix. Then the system of linear equations $(A^{2}B^{2} - B^{2}A^{2})X = O$,where $X$ is a $3 \times 1$ column matrix of unknown variables and $O$ is a $3 \times 1$ null matrix,has ....... .

If $A$ and $B$ are square matrices of order $3$,then $|(A-A^T)+(B-B^T)|=$

If $A$ is a square matrix such that $A^{2} = A$,then $(I + A)^{3} - 7A$ is equal to