Consider the mass$-$spring system shown below.

The frequencies for the mass vibrations can be determined by solving for the eigenvalues and by applying $M{x}^{}-kx=0,$ which yields

$\left[\begin{array}{ccc}{m}_1& 0& 0\\ 0& m_2& 0\\ 0& 0& m_3\end{array}\right]\{[x_1^{}],[x_2^{}],[x_3^{}]\}+\left[\begin{array}{ccc}2k& -k& -k\\ -k& 2k& -k\\ -k& -k& 2k\end{array}\right]\{[x_1],[x_2],[x_3]\}=\{[0],[0],[0]\}$

Applying the guess $x=x_0{e}^{it}$ as a solution, we get the following matrix:

$\left[\begin{array}{ccc}2k-m_1{}^2& -k& -k\\ -k& 2k-m_2{}^2& -k\\ -k& -k& 2k-m_3{}^2\end{array}\right]\{[x_{01}],[x_{02}],[x_{03}]\}{e}^{it}=\{[0],[0],[0]\}$

Use MATLAB's "eig" command to solve for the eigenvalues of the $k-m{}^2$ matrix above. Also, solve for the eigenvalues and report the frequencies $().$

Let $m_1=m_2=m_3=1kg$ and $k=2\frac{N}{m}.$