###### Exercise 3

Suppose that \(A\) is an \(n\times n\) matrix.

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

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

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 \(\lambda^{-1}\text{.}\)

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 \(\lambda^2\text{.}\)

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 \(A^5\text{?}\)