The Jordan canonical form of the matrix $A$ is \[ J = \left[ \begin{array}{ccc} J_1 & & \\ & J_2 & \\ & & J_3 \end{array} \right] \] where the Jordan blocks $J_1$, $J_2$, and $J_3$ have sizes $1 \times 1$, $2 \times 2$, and $3 \times 3$, respectively, all associated with the same eigenvalue $\lambda$.
What is the rank of $\lambda I-A$?