Row Reduction & Rank · Superchallenge problem

For a real number $a$, let $r(a)$ be the rank of the $4 \times 4$ matrix with every diagonal entry equal to $a$ and every off-diagonal entry equal to $1$. Compute $\displaystyle\sum_{a \in \mathbb R} \bigl(4 - r(a)\bigr)$ (only finitely many terms are nonzero).