Matrix $A$ satisfies $A^2 = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,$A^n$ is equal to $(n \in N)$:

The number of matrices $A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$,where $a, b, c, d \in \{-1, 0, 1, 2, 3, \ldots, 10\}$,such that $A=A^{-1}$,is

If $A$ and $B$ are two square matrices of the same order such that $AB = B$ and $BA = A$,then $A^{2} + B^{2}$ is always equal to

Let $A = \begin{bmatrix} a_1 \\ a_2 \end{bmatrix}$ and $B = \begin{bmatrix} b_1 \\ b_2 \end{bmatrix}$ be two $2 \times 1$ matrices with real entries such that $A = XB$,where $X = \frac{1}{\sqrt{3}} \begin{bmatrix} 1 & -1 \\ 1 & k \end{bmatrix}$ and $k \in R$. If $a_1^2 + a_2^2 = \frac{2}{3}(b_1^2 + b_2^2)$ and $(k^2 + 1)b_2^2 \neq -2b_1b_2$,then the value of $k$ is ....... .

$A=\begin{bmatrix} 1 & 0 & 1 \\ 0 & 1 & 1 \\ 0 & 1 & 0 \end{bmatrix} \Rightarrow A^2-2A=$

For any $3 \times 3$ matrix $M$,let $| M |$ denote the determinant of $M$. Let $E=\begin{bmatrix} 1 & 2 & 3 \\ 2 & 3 & 4 \\ 8 & 13 & 18 \end{bmatrix}$,$P=\begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$ and $F=\begin{bmatrix} 1 & 3 & 2 \\ 8 & 18 & 13 \\ 2 & 4 & 3 \end{bmatrix}$. If $Q$ is a nonsingular matrix of order $3 \times 3$,then which of the following statements is (are) $TRUE$?
$(A)$ $F = PEP$ and $P^2 = I$
$(B)$ $| EQ + PFQ^{-1} | = | EQ | + | PFQ^{-1} |$
$(C)$ $|(EF)^3| > |EF|^2$
$(D)$ The sum of the diagonal entries of $P^{-1}EP + F$ is equal to the sum of the diagonal entries of $E + P^{-1}FP$