$M$:$6\mathrm{.}72$ Consider the eigenvalue problem:

${}^2\left[\begin{array}{cc}1& 0\\ 0& 2\end{array}\right]\{[x_1],[x_2]\}=\left[\begin{array}{cc}6& -2\\ -2& 2\end{array}\right]\{[x_1],[x_2]\}$

a$.$ Find the natural frequencies and mode shapes of the system.

b$.$ Change the coordinates in Eq$.($E$\mathrm{.}1)$ as $x_1=Y_1$ and $x_2=3Y_2$ and express the eigenvalue

problem in terms of the eigenvector vec$(Y)=\{[Y_1],[Y_2]\},$ solve it$,$ and find the natural frequencies

and mode shapes.

c$.$ Compare the results found in parts $($a$)$ and $($b$)$ and give your observations.