###### Exercise3

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

1. Explain why $$\lambda = 0$$ is an eigenvalue if and only if there is a non-trivial solution to the homogeneous equation $$A\xvec = 0\text{.}$$

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

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

4. If $$A$$ is an $$n\times n$$ matrix with eigenvector $$\vvec$$ and associated eigenvalue $$\lambda\text{,}$$ explain why $$\vvec$$ is also an eigenvector of $$A^2$$ with associated eigenvalue $$\lambda^2\text{.}$$

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

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