There really is no need to calculate the characteristic polynomial that way.
First, you can easily observe that 0 is an eigenvalue of geometric multiplicty 1 (since . Now, note that and A only has two eigenvalues, so the second one must be 2. Another way to know that 2 is an eigenvalue is to note that the sum of all rows is 2, therefore, according to a theorem which you should have learned, 2 is an eigenvalue with corresponding eigenvector .
So we know that 2 is an eigenvalue of geometric multiplicity and 0 is an eigenvector with geometric multiplicity 1. Since algebraic multiplicity geometric multiplicity for each eigenvalue, we get that the algebraic and geometric multiplicities for each eigenvector are 1 and 1.
With 0,2 as eigenvalues
and the corresponding eigenvectors.
However, if you still want to know what you did wrong --