Exercise 7

Determine whether the following statements are true or false and provide a justification for your response.

1. The eigenvalues of a diagonal matrix are equal to the entries on the diagonal.

2. If $A\vvec=\lambda\vvec$, then $A^2\vvec=\lambda\vvec$ as well.

3. Every vector is an eigenvector of the identity matrix.

4. If $\lambda=0$ is an eigenvalue of $A$, then $A$ is invertible.

5. For every $n\times n$ matrix $A$, it is possible to find a basis of $\real^n$ consisting of eigenvectors of $A$.