Let $A=\left[\begin{array}{lll}x & y & z \\ y & z & x \\ z & x & y\end{array}\right], \quad$ where $x, y$ and $z$ are real numbers such that $x + y + z >0$ and $xyz =2$ If $A ^{2}= I _{3},$ then the value of $x ^{3}+ y ^{3}+ z ^{3}$ is …………

If $A^{\prime}=\left[\begin{array}{cc}3 & 4 \\ -1 & 2 \\ 0 & 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}-1 & 2 & 1 \\ 1 & 2 & 3\end{array}\right],$ then verify that $(A-B)^{\prime}=A^{\prime}-B^{\prime}$

Show that the matrix $A=\left[\begin{array}{ccc}1 & -1 & 5 \\ -1 & 2 & 1 \\ 5 & 1 & 3\end{array}\right]$ is a symmetric matrix

The set of all values of $t \in R$, for which the matrix 

$\left[\begin{array}{ccc}e^t & e^{-t}(\sin t-2 \cos t) & e^{-t}(-2 \sin t-\cos t) \\e^t & e^{-t}(2 \sin t+\cos t) & e^{-t}(\sin t-2 \cos t) \\e^t & e^{-t} \cos t & e^{-t} \sin t \end{array}\right]$ is invertible.

If $A$ and $B$ are symmetric matrices of the same order, then show that $AB$ is symmetric if and only if $A$ and $B$ commute, that is $AB = BA$.