If $D_1$ and $D_2$ are two $3 \times 3$ diagonal matrices,then

Let $A, B, C$ be $3 \times 3$ non-singular matrices and $I$ be the identity matrix of order three. If $A B A = B A^2 B$ and $A^3 = I$,then $A B^4 - B^4 A = $

Let $A$ and $B$ be two non-singular matrices of order $3$ such that $A + B = I$ and $A^{-1} + B^{-1} = 2I$. Then $|adj(4AB)|$ is equal to (where $adj(A)$ is the adjoint of matrix $A$):

Let $|M|$ denote the determinant of a square matrix $M$. Let $g:\left[0, \frac{\pi}{2}\right] \rightarrow R$ be the function defined by $g(\theta)=\sqrt{f(\theta)-1}+\sqrt{f\left(\frac{\pi}{2}-\theta\right)-1}$,where $f(\theta)=\frac{1}{2}\left|\begin{array}{ccc}1 & \sin \theta & 1 \\ -\sin \theta & 1 & \sin \theta \\ -1 & -\sin \theta & 1\end{array}\right|+\left|\begin{array}{ccc}\sin \pi & \cos \left(\theta+\frac{\pi}{4}\right) & \tan \left(\theta-\frac{\pi}{4}\right) \\ \sin \left(\theta-\frac{\pi}{4}\right) & -\cos \frac{\pi}{2} & \log _e\left(\frac{4}{\pi}\right) \\ \cot \left(\theta+\frac{\pi}{4}\right) & \log _e\left(\frac{\pi}{4}\right) & \tan \pi\end{array}\right|$. Let $p(x)$ be a quadratic polynomial whose roots are the maximum and minimum values of the function $g(\theta)$,and $p(2)=2-\sqrt{2}$. Then,which of the following is/are $TRUE$?
$(A) \ p \left(\frac{3+\sqrt{2}}{4}\right) < 0$
$(B) \ p \left(\frac{1+3 \sqrt{2}}{4}\right)>0$
$(C) \ p \left(\frac{5 \sqrt{2}-1}{4}\right)>0$
$(D) \ p \left(\frac{5-\sqrt{2}}{4}\right) < 0$

Let $B=\left[\begin{array}{ll}1 & 2 \\ 0 & 1\end{array}\right]$ and $A$ be a $2 \times 2$ matrix satisfying $\left(A^T\right)^{-1}=A$. If $X=A B A^T$,then $A^T X^{2021} A=$

Let $a$ and $b$ be non-zero real numbers such that $ab = 5/2$. Given $A = \begin{bmatrix} a & -b \\ b & a \end{bmatrix}$ and $AA^T = 20I$ (where $I$ is the identity matrix),the quadratic equation whose roots are $a$ and $b$ is: