The minimal polynomial $m_A(\lambda)$ of a matrix $A$ is the monic polynomial of smallest degree such that $m_A(A) = 0$. It divides the characteristic polynomial and has the same roots but with multiplicities equal to the size of the largest Jordan block for each eigenvalue. For a $3 \times 3$ matrix with characteristic polynomial $p(\lambda) = (\lambda - 2)^3$ that has exactly 2 linearly independent eigenvectors for $\lambda = 2$, what is the minimal polynomial?