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

Matrix $A$ is such that ${A^2} = 2A - I$,where $I$ is the identity matrix. Then for $n \ge 2$,${A^n} = $

Let $A = [a_{ij}]_{2 \times 2}$ where $a_{ij} \neq 0$ for all $i, j$ and $A^2 = I$. Let $a$ be the sum of all diagonal elements of $A$ and $b = |A|$. Then $3a^2 + 4b^2$ is equal to:

Let $A = \begin{bmatrix} \cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{bmatrix}$,$\alpha \in R$ such that $A^{32} = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}$. Then a value of $\alpha$ is

If $A = \begin{bmatrix} \cos^2 \theta & \sin \theta \cos \theta \\ \sin \theta \cos \theta & \sin^2 \theta \end{bmatrix}$,$B = \begin{bmatrix} \cos^2 \phi & \sin \phi \cos \phi \\ \sin \phi \cos \phi & \sin^2 \phi \end{bmatrix}$ and $\theta$ and $\phi$ differ by $\frac{\pi}{2}$,then $AB = $

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).