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

If $A$ and $B$ are $3 \times 3$ matrices such that $AB = A$ and $BA = B$,then

Let $S=\{n \in N \mid \begin{bmatrix} 0 & i \\ 1 & 0 \end{bmatrix}^{n} \begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \forall a, b, c, d \in R \}$,where $i=\sqrt{-1}$. Then the number of $2$-digit numbers in the set $S$ is $......$

If $A = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$ and $B = \begin{bmatrix} 0 & -i \\ i & 0 \end{bmatrix}$,then $(A + B)^2$ equals

Let $A$ be the set of all $3 \times 3$ matrices with entries $0$ or $1$ only. Let $B$ be the subset of $A$ consisting of all matrices with determinant value $1$. Let $C$ be the subset of $A$ consisting of all matrices with determinant value $-1$. Then:

If $A = \left\{ \begin{bmatrix} a & b \\ c & d \end{bmatrix} : a, b, c, d \in \{-1, 1\} \right\}$,then the number of singular matrices in $A$ is