If $P$ and $Q$ are two non-singular matrices of the same order such that $Q^r = I$,for some integer $r > 1$,then $P^{-1}Q^{r-1}P - P^{-1}Q^{-1}P$ is equal to (where $I$ is the identity matrix and $O$ is the null matrix).

If $A$ and $B$ are two invertible matrices of the same order,then $adj \,(AB)$ is equal to :-

Let $A = \begin{bmatrix} 2 & -5 \\ 3 & 1 \end{bmatrix}$. What is $f(A)$ if $f(x) = x^3 - 2x^2 - 5$?

Let $A = (a_{ij})$ be an $n \times n$ matrix defined by $a_{ij} = \begin{cases} k^i, & \forall i=j \\ 0, & \text{otherwise} \end{cases}$. If $m = \text{trace of } A$ and $\lim_{k \rightarrow 1} \frac{n-m}{1-k} = 171$, then the value of $n$ is:

Let $A$ be a square matrix of order $2$ such that $|A|=2$ and the sum of its diagonal elements is $-3$. If the points $(x, y)$ satisfying $A^2+xA+yI=0$ lie on a hyperbola,whose transverse axis is parallel to the $x$-axis,eccentricity is $e$ and the length of the latus rectum is $\ell$,then $e^4+\ell^4$ is equal to?

Let $A$ be a $2 \times 2$ matrix with real entries such that $A^{T} = \alpha A + I$,where $\alpha \in R - \{-1, 1\}$. If $\det(A^2 - A) = 4$,then the sum of all possible values of $\alpha$ is equal to