###### Exercise4

Provide a justification for your response to the following questions.

1. Suppose that $$A$$ is a $$3\times 3$$ matrix having eigenvalues $$\lambda = -3,3,-5\text{.}$$ What are the eigenvalues of $$2A\text{?}$$

2. 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 $$D\text{?}$$

3. 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 $$A\text{?}$$

4. 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 $$A\text{?}$$

5. 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 $$A^{-1}\text{?}$$

in-context