A stress tensor $$ has a matrix $$ given below.

$[]=\left[\begin{array}{ccc}5& 4& 0\\ 4& -1& 0\\ 0& 0& 3\end{array}\right]$

$($a$)$ Write the characteristic equation and find the principal stresses and their corresponding

principal directions. $($b$)$ Find the principal scalar stress invariants. $($c$)$ If $n_1,n_2,n_3$ are the

principal directions, write $[{]}_{n_j}.($d$)$ Could the following matrix $S$ represent the same tensor

$$ with respect to some basis? Please write it out.

$[S]=\left[\begin{array}{ccc}7& 2& 0\\ 2& 1& 0\\ 0& 0& -1\end{array}\right]$