Exercise 3

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

1. Explain why $\lambda = 0$ is an eigenvalue of $A$ if and only if there is a nonzero solution to the homogeneous equation $A\xvec = 0$.

2. Explain why $A$ is not invertible if and only if $\lambda=0$ is an eigenvalue.

3. If $\vvec$ is an eigenvector of $A$ having associated eigenvalue $\lambda$, explain why $\vvec$ is also an eigenvector of $A^2$ with associated eigenvalue $\lambda^2$.

4. If $A$ is invertible and $\vvec$ is eigenvector of $A$ having associated eigenvalue $\lambda$, explain why $\vvec$ is also an eigenvector of $A^{-1}$ with associated eigenvalue $\lambda^{-1}$.

5. The matrix $A=\ \left[\begin{array}{rr} 1 & 2 \\ 2 & 1 \\ \end{array}\right]$ has eigenvectors $\vvec_1=\twovec{1}{1}$ and $\vvec_2=\twovec{-1}{1}$ and associated eigenvalues $\lambda_1 = 3$ and $\lambda=-1$. What are some eigenvectors and associated eigenvalues for $A^5$?