Exercise 4
Provide a justification for your response to the following questions.
Suppose that \(A\) is a \(3\times 3\) matrix having eigenvalues \(\lambda = 3,3,5\text{.}\) What are the eigenvalues of \(2A\text{?}\)
Suppose that \(D\) is a diagonal \(3\times 3\) matrix. Why can you guarantee that there is a basis of \(\real^3\) consisting of eigenvectors of \(D\text{?}\)
If \(A\) is a \(3\times 3\) matrix whose eigenvalues are \(\lambda = 1,3,5\text{,}\) can you guarantee that there is a basis of \(\real^3\) consisting of eigenvectors of \(A\text{?}\)

Suppose that the characteristic polynomial of a matrix \(A\) is
\begin{equation*} \det(A\lambda I) = \lambda^3 + 4\lambda\text{.} \end{equation*}What are the eigenvalues of \(A\text{?}\) Is \(A\) invertible? Is there a basis of \(\real^n\) consisting of eigenvectors of \(A\text{?}\)

If the characteristic polynomial of \(A\) is
\begin{equation*} \det(A\lambda I) = (4 \lambda)(2\lambda)(1\lambda)\text{,} \end{equation*}what is the characteristic polynomial of \(A^2\text{?}\) what is the characteristic polynomial of \(A^{1}\text{?}\)