Exercise 8

Suppose that $A$ is an $n\times n$ matrix.

1. Assuming that $\vvec$ is an eigenvector of $A$ whose associated eigenvalue is nonzero, explain why $\vvec$ is in $\col(A)$.

2. Assuming that $\vvec$ is an eigenvector of $A$ whose associated eigenvalue is zero, explain why $\vvec$ is in $\nul(A)$.

3. Consider the two special matrices below and find their eigenvectors and associated eigenvalues. $$A = \left[\begin{array}{rrr} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \\ \end{array}\right],~~~ B = \left[\begin{array}{rrr} 1 & 1 & 1 \\ 2 & 2 & 2 \\ 3 & 3 & 3 \\ \end{array}\right].$$