For what value of $x\in[0,\frac{\pi}{2}]$, is $A+A'=\sqrt{3}I$, where $A=\begin{bmatrix}\cos x & \sin x\\ -\sin x & \cos x\end{bmatrix}$ ?

Four friends Abhay, Bina, Chhaya and Devesh were asked to simplify \(4~AB+3(AB+BA)-4~BA,\) where A and B are both matrices of order \(2\times2\). It is known that \(A\ne B\ne I\) and \(A^{-1}\ne B\). Their answers are given as: Abhay: \(6 AB\), Bina : \(7 AB-BA\), Chhaya: \(8 AB\), Devesh: \(7 BA - AB\). Who answered it correctly?

Which of the following can be both a symmetric and skew-symmetric matrix ?

If A and B are square matrices of order m such that \(A^{2}-B^{2}=(A-B)(A+B),\) then which of the following is always correct?

If the sum of all the elements of a \(3\times3\) scalar matrix is 9, then the product of all its elements is: