1. Hint: If the eigenvalues of A are all nonzero then what can you conclude about the value of det(A)?

5. Hint: Whether or not the matrix is defective depends upon the number of linearly independent eigenvectors.

6. Hint: The eigenspace corresponding to $\lambda =0$ is N(A), so the dimension of the eigenspace is equal to the nullity of A.

7. Hint: The hint for question 6 also applies to this question.

8. Hint: Look at some triangular matrices that have some 0’s on the diagonal.

9. Hint: See the list of observations following the proof of Theorem6.5.1

11. Hint: $U^{-1}=U^H,\text{so}A=UT{U}^{-1}\mathrm{.}$

12. Hint: If A is normal, then A can be factored into a product $A=UD{U}^H$ where U is unitary and D is diagonal.

13. Hint: What do we know about the eigenvalues of A and $A^{-1}?$

14. Hint: Look at some examples of $2\times 2$ diagonal matrices.