Exercise 1.6.12

There are algebraic multiplicity of 2, where ${\lambda }_{1,2}=0$ and ${\lambda }_3=6$.

We have eigenvectors from $\mathit{Ax}=\mathit{\lambda x}$: * When $\lambda =6$, the eigenvector is $v_1=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 2\hfill \\ \hfill 1\hfill \end{array}\right]$ * When $\lambda =0$, we have $\mathit{Ax}=0$, so the eigenvectors are in the nullspace of $A$, e.g. $v_2=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill -1\hfill \\ \hfill -1\hfill \end{array}\right]$ and $v_3=\left[\begin{array}{c}\hfill 1\hfill \\ \hfill 1\hfill \\ \hfill -3\hfill \end{array}\right]$

There are two independent eigenvectors for $\lambda =0$, so the geometric multiplicity = 2 as well.