Consider the eigenvalue problem [left[[k]-omega^{2}[m] ight] vec{X}=overrightarrow{0}] where [[m]=left[begin{array}{cc}2 & 0 0 & 1end{array} ight] text {

Consider the eigenvalue problem

\[\left[[k]-\omega^{2}[m]\right] \vec{X}=\overrightarrow{0}\]

where

\[[m]=\left[\begin{array}{cc}2 & 0 \\0 & 1\end{array}\right] \text { and }[k]=\left[\begin{array}{cc}8 & -4 \\-4 & 4\end{array}\right]\]

Find the natural frequencies and mode shapes of the system:

a. by solving the equation

\[\left[[m]^{-1}[k]-\omega^{2}[I]\right] \vec{X}=\overrightarrow{0}\]

b. by solving the equation

\[\left[-\omega^{2}[k]^{-1}[m]+[I]\right] \vec{X}=\overrightarrow{0}\]

c. Compare the two sets of results and give your observations.