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

The number of $3 \times 3$ non-singular matrices,with four entries as $1$ and all other entries as $0$,is

Let $A = \begin{bmatrix} 1 & a & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix}$,where $a, b \in \mathbb{R}$. If for some $n \in \mathbb{N}$,$A^n = \begin{bmatrix} 1 & 48 & 2160 \\ 0 & 1 & 96 \\ 0 & 0 & 1 \end{bmatrix}$,then $n + a + b$ is equal to:

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

If $\Delta_{r}=\left|\begin{array}{cc}\frac{1}{3r-2} & \frac{2}{3r-5} \\ 0 & \frac{3}{3r+1}\end{array}\right|$, then $\sum_{r=1}^{33} \Delta_{r}=$

Among the statements:
$I$: If $\begin{vmatrix} 1 & \cos \alpha & \cos \beta \\ \cos \alpha & 1 & \cos \gamma \\ \cos \beta & \cos \gamma & 1 \end{vmatrix} = \begin{vmatrix} 0 & \cos \alpha & \cos \beta \\ \cos \alpha & 0 & \cos \gamma \\ \cos \beta & \cos \gamma & 0 \end{vmatrix}$, then $\cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma=\frac{3}{2}$
$II$: If $\begin{vmatrix} x^{2}+x & x+1 & x-2 \\ 2x^{2}+3x-1 & 3x & 3x-3 \\ x^{2}+2x+3 & 2x-1 & 2x-1 \end{vmatrix} = px+q$, then $p^{2}=196q^{2}$