An nxn matrix A has an inverse
   <=> A^(-1) exists
   <=> rref(A) = I
   <=> A is invertible
   <=> A is non-singular
   <=> 0 is not an eigenvalue for A
   <=> det(A) not= 0 

An nxn matrix A does not have inverse
   <=> A^(-1) does not exist
   <=> rref(A) not= I
   <=> A is not invertible
   <=> A is singular
   <=> 0 is an eigenvalue for A
   <=> det(A) = 0 

The characteristic polynomial of the nxn matrix A is p(s) = det(A-sI)

The scalar r is an eigenvalue for the nxn matrix A <=> r is a root
of the characteristic polynomial p(s), that is p(r) = 0.

Complex numbers can be eigenvectors. The matrices in this class are 
carefully chosen so the only eigenvalues are real. For example, symmetric
matrices have only real eigenvalues.

If X1 and X2 are eigenvectors so that A*X1 = L1*X1 and A*X2 = L2*X2 then
for any X = C1*X1 + C2*X2 you can compute A*X by
    A*X = A(C1*X1 + C2*X2) = C1*A*X1 + C2*A*X2 = C1*L1*X1 + C2*L2*X2