If $\omega$ is a root of the equation $x+\frac{1}{x}+1=0$,then the value of the determinant $\left|\begin{array}{ccc}1 & 1+\omega & 1+\omega+\omega^2 \\ 3 & 4+3 \omega & 5+4 \omega+3 \omega^2 \\ 6 & 9+6 \omega & 11+9 \omega+6 \omega^2\end{array}\right|$ is equal to

If $AB = A$ and $BA = B$,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 ....... .

Let $A, B$ be two $3 \times 3$ matrices and $C$ be a $3 \times 3$ identity matrix such that $AB-C$ is a non-singular matrix. Let $D=(AB-C)^{-1}$. Then,consider the following statements.
Statement $I$: $\operatorname{det}(BA)=\operatorname{det}(BA-C) \operatorname{det}(BDA)$
Statement $II$: $ABD=DAB$
Which of the above statements is (are) true?

If $P$ is a non-singular matrix such that $I+P+P^2+\ldots+P^{n}=0$ ($0$ denotes the null matrix),then $P^{-1}=$

Let $A = \begin{bmatrix} 1+i & 1 \\ -i & 0 \end{bmatrix}$ where $i = \sqrt{-1}$. Then,the number of elements in the set $\{n \in \{1, 2, \ldots, 100\} : A^n = A\}$ is