Let $p$ be a non-singular matrix such that $I + p + p^2 + .... + p^n = O$ (where $O$ denotes the null matrix and $I$ denotes the identity matrix),then $p^{-1} = $

Let $P=\begin{bmatrix} -30 & 20 & 56 \\ 90 & 140 & 112 \\ 120 & 60 & 14 \end{bmatrix}$ and $A=\begin{bmatrix} 2 & 7 & \omega^{2} \\ -1 & -\omega & 1 \\ 0 & -\omega & -\omega+1 \end{bmatrix}$,where $\omega=\frac{-1+ i \sqrt{3}}{2}$,and $I_{3}$ is the identity matrix of order $3$. If the determinant of the matrix $(P^{-1}AP - I_{3})^{2}$ is $\alpha \omega^{2}$,then the value of $\alpha$ is equal to:

If $A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & k & 2 \\ 4 & 1 & 5 \end{bmatrix}$ is a singular matrix,then the quadratic equation having the roots $k$ and $\frac{1}{k}$ is

If $A = \begin{bmatrix} 3 & 4 \\ 5 & 6 \end{bmatrix}$ and $B = \begin{bmatrix} x & 0 \\ 0 & y \end{bmatrix}$,where $x, y \in \mathbb{N}$,then:

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

If $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$ satisfies the equation $x^2 - (a + d)x + k = 0$,then