Algebraic and geometric multiplicities of eigenvalues of a $3 \times 3$ matrix "Let A = \begin{pmatrix}-1&-1&-2\\ \:2&2&1\\ \:6&2&6\end{pmatrix}
Find the characteristic polynomial of A. 
Find the distinct eigenvalues of A and their respective algebraic and geometric multiplicities."
So I calculated the value of the characteristic polynomial and got x^3 -7x^2 + 16 x -12 or (x - 3)(x - 2)(x - 2) and got the eigenvalues of 2 and 3. I believe these are the correct answers. However, I am not sure how to give the geometric and algebraic multiplicities for the eigenvalues. I thought that for the eigenvalue 3 the multiplicity would have been 1 and for the eigenvalue 2 it would have been 2. But it seems like I need two multiplicity values for each.
Any help?

The numbers you are quoting are the algebraic multiplicity; i.e., $\lambda$ has algebraic multiplicity $n$ when the factor $(x - \lambda)$ appears exactly $n$ times in the factored characteristic polynomial. 
The geometric multiplicity of an eigenvalue $\lambda$ is the dimension of the eigenspace corresponding to $\lambda$. That is, what is the size of the largest set of linearly independent eigenvectors you can create for $\lambda$.
