Definition: Eigenvalues of a matrix 1) Can a non-square matrix have eigenvalues? Why?
2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible. 
Thank you!
 A: For 1) No, it has to be a square matrix by definition.
To see why, consider the following:
Recall that for an eigenvector $v$ and and an eigenvalue $\lambda$, you have that $Av$ = $\lambda v$. 
Now suppose that dim(v) = n x 1. That means that dim(Av) = n x 1 and dim($\lambda v$) = n x 1. If A is not square then dim(A) must be m x n where $m \neq n$. But then you have that dim($Av$) = (m x n) * (n x1) = m x 1. But we just said that dim(Av) = n x 1 . Thus contradiction.
In regards to your second question, this post answers it completely.
Is a matrix with characteristic polynomial $t^2 +1$ invertible?
A: Hint.  (1) Try it!  For example, can you solve
$$\pmatrix{1&2&3\cr4&5&6\cr}\pmatrix{v_1\cr v_2\cr v_3\cr}
  =\lambda\pmatrix{v_1\cr v_2\cr v_3\cr}\ ?$$
(2) You should know


*

*the connection between whether or not $A$ is invertible and $\det(A)$;

*the connection between $\det(A)$ and the eigenvalues of $A$;

*the connection between the eigenvalues of $A$ and its characteristic polynomial.

