prove the following results for the eigenvalues of an $n \times n$ matrix $A$ Prove the following results for the eigenvalues of an $n \times  n$ matrix $A$: 
(a) $0$ is an eigenvalue for $A$ if and only if $A$ is not invertible.
(b) $A$ and $A^T$ have the same eigenvalues.

(a) Suppose $0$ is an eigenvalue of $A$. Then $Av=0v$ for some
nonzero vector $v$. What does that mean?
(b) Look at the characteristic polynomial of the matrix $A$ and the
characteristic polynomial of its transpose. What can you say about
them?
