Exercise 4

Provide a justification for your response to the following questions.

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

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 $D$?

3. If $A$ is a $3\times 3$ matrix whose eigenvalues are $\lambda = -1,3,5$, can you guarantee that there is a basis of $\real^3$ consisting of eigenvectors of $A$?

4. Suppose that the characteristic polynomial of a matrix $A$ is $$\det(A-\lambda I) = -\lambda^3 + 4\lambda$$. What are the eigenvalues of $A$? Is $A$ invertible? Is there a basis of $\real^n$ consisting of eigenvectors of $A$?

5. If the characteristic polynomial of $A$ is $$\det(A-\lambda I) = (4 -\lambda)(-2-\lambda)(1-\lambda)$$, what is the characteristic polynomial of $A^2$? what is the characteristic polynomial of $A^{-1}$?