Consider a system as follows

$\left[\begin{array}{cc}6& 2\\ 2& 4\end{array}\right]x^{}(t)+\left[\begin{array}{cc}3& -1\\ -1& 1\end{array}\right]x(t)=0$

$($a$)$ Calculate the characteristic equation and solve for the system's natural frequencies.

$($b$)$ Find mode shape vectors $u_1$ and $u_2.$