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

Let the minimum $m$ $(m \in Z^+)$ be defined as the power of a square matrix $A$ such that $A^m = I$. If $A^5 = I$ and $ABA^{-1} = B^2$,then the power of matrix $B$ such that $B^k = I$ is between:

Let $\omega \neq 1$ be a cube root of unity and $S$ be the set of all non-singular matrices of the form $\begin{bmatrix} 1 & a & b \\ \omega & 1 & c \\ \omega^2 & \omega & 1 \end{bmatrix}$ where each of $a, b$,and $c$ is either $\omega$ or $\omega^2$. Then the number of distinct matrices in the set $S$ is

If $A$ and $B$ are two invertible square matrices of the same order such that $(A + B)(A - B) = A^2 - B^2$,then $(A^2BA^{-1}B^{-1})^3$ is equal to-

Which of the following statements is correct about two square matrices $A$ and $B$ of the same order $n$?

Let $A$ be a $2 \times 2$ matrix with real entries. Let $I$ be the $2 \times 2$ identity matrix. Denote by $tr(A)$ the sum of diagonal entries of $A$. Assume that $A^2 = I$.
Statement-$1$: If $A \neq I$ and $A \neq -I$,then $\det(A) = -1$.
Statement-$2$: If $A \neq I$ and $A \neq -I$,then $tr(A) \neq 0$.