Let $A = \begin{bmatrix} 0 & 1 & 2 \\ a & 0 & 3 \\ 1 & c & 0 \end{bmatrix}$,where $a, c \in \mathbb{R}$. If $A^3 = A$ and the positive value of $a$ belongs to the interval $(n-1, n]$,where $n \in \mathbb{N}$,then $n$ is equal to $..........$.

If ${a^2} + {b^2} + {c^2} = -2$ and $f(x) = \left| {\begin{array}{*{20}{c}}{1 + {a^2}x}&{(1 + {b^2})x}&{(1 + {c^2})x}\\{(1 + {a^2})x}&{1 + {b^2}x}&{(1 + {c^2})x}\\{(1 + {a^2})x}&{(1 + {b^2})x}&{1 + {c^2}x}\end{array}} \right|$,then $f(x)$ is a polynomial of degree:

Consider a matrix $A = \begin{bmatrix} \alpha & \beta & \gamma \\ \alpha^{2} & \beta^{2} & \gamma^{2} \\ \beta+\gamma & \gamma+\alpha & \alpha+\beta \end{bmatrix}$,where $\alpha, \beta, \gamma$ are three distinct natural numbers. If $\frac{\operatorname{det}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj}(\operatorname{adj} A))))}{(\alpha-\beta)^{16}(\beta-\gamma)^{16}(\gamma-\alpha)^{16}}=2^{32} \times 3^{16}$,then the number of such $3$-tuples $(\alpha, \beta, \gamma)$ 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-

Let $A, B, C, D$ be square real matrices such that $C^T = DAB$,$D^T = ABC$,and $S = ABCD$. Then $S^2$ is equal to:

If $A=\left[\begin{array}{ccc}2 & 0 & -3 \\ 4 & 3 & 1 \\ -5 & 7 & 2\end{array}\right]$ is expressed as a sum of a symmetric matrix $P$ and a skew-symmetric matrix $Q$,then $P^{T}-Q^{T}=$