Let $I$ be an identity matrix of order $2 \times 2$ and $P = \begin{bmatrix} 2 & -1 \\ 5 & -3 \end{bmatrix}$. Then the value of $n \in N$ for which $P^n = 5I - 8P$ is equal to ..... .

If $A = \begin{bmatrix} -4 & -1 \\ 3 & 1 \end{bmatrix}$,then the determinant of the matrix $(A^{2016} - 2A^{2015} - A^{2014})$ is

If $\omega (\neq 1)$ is a cube root of unity,then the value of the determinant $\left| \begin{array}{ccc} 1 & 1 + i + \omega^2 & \omega^2 \\ 1 - i & -1 & \omega^2 - 1 \\ -i & -i + \omega - 1 & -1 \end{array} \right|$ is equal to

Read the following mathematical statements carefully:
$I$. There can exist two triangles such that the sides of one triangle are all less than $1 \text{ cm}$ while the sides of the other triangle are all bigger than $10 \text{ m}$,but the area of the first triangle is larger than the area of the second triangle.
$II$. If $x, y, z$ are all different real numbers,then $\frac{1}{(x - y)^2} + \frac{1}{(y - z)^2} + \frac{1}{(z - x)^2} = \left( \frac{1}{x - y} + \frac{1}{y - z} + \frac{1}{z - x} \right)^2$.
$III$. $\log_3 x \cdot \log_4 x \cdot \log_5 x = (\log_3 x \cdot \log_4 x) + (\log_4 x \cdot \log_5 x) + (\log_5 x \cdot \log_3 x)$ is true for exactly one real value of $x$.
$IV$. $A$ matrix has $12$ elements. The number of possible orders it can have is $6$. Now indicate the correct alternative.

Let $A = \begin{bmatrix} 2 & 1 & 2 \\ 6 & 2 & 11 \\ 3 & 3 & 2 \end{bmatrix}$ and $P = \begin{bmatrix} 1 & 2 & 0 \\ 5 & 0 & 2 \\ 7 & 1 & 5 \end{bmatrix}$. The sum of the prime factors of $|P^{-1}AP - 2I|$ is equal to

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 $.....$