Problem $2.$ A second order tensor A in the vector basis $\{e_i\}$ is given by

$A_{ij}=-2{}_{i1}{}_{j1}-2{}_{i2}{}_{j2}+{}_{i3}{}_{j3}+{}_{i1}{}_{j2}+{}_{i2}{}_{j1}$

$($a$)$ Show that $A_{ij}=A_{ji}$ and that the matrix form of $A$ is$.$

$A=\left[\begin{array}{ccc}-2& 1& 0\\ 1& -2& 0\\ 0& 0& 1\end{array}\right]$

$($b$)$ Determine the components of $A$ in the vector basis $\{e_i\}$ obtained by rotating $\{e_i\}$ through

$45$ about $e_3.$

$($c$)$ What is special about the result in $($b$)?$