Calculating eigenvalues and eigenvectors of matrices by hand can be a daunting task. This is why homework problems deal mostly with 2x2 or 3x3 matrices. For 2x2, 3x3, and 4x4 matrices, there are complete answers to the problem. In that case, one can give explicit algebraic formulas for the solutions. For 5x5 matrices, an explicit algebraic solution can not be found any more since one would have to give formulas of the roots of a polynomial of degree 5. Below, we have the solution for the 2x2 case. Your linear algebra teacher probably doesn't want you to know them... 
Let T=a+d be the trace and D=adbc be the determinant of
the matrix

The eigenvalues of A are


If c is not zero, then the eigenvectors are

If b is not zero, then the eigenvectors are

If both b and c are zero, then the eigenvectors are

Proof. We just have to verify that A v = L v holds. In all cases, one of the two equations is (bcad) + (a+d) L  L^{2}=0 and in the other equation, everything cancels. 