If $P \in M_7(\mathbb{R})$ has rank 4, rank of $P + aa^T$, where $a$ is a column vector?

Question

Let $P$ be a $7 \times 7$ matrix of rank $4$ with real entries and let $a \in \mathbb{R}^7$ be a column vector. Then the rank of $P + aa^T$ is at least ______.

This question was in the 2017 IIT JAM paper. Any ideas?

Answer 1

HINT.

• Use the inequality $\text{Rank}(A+B)\le \text{Rank}(A)+\text{Rank}(B)$ to show that

$\qquad\text{Rank}(A+B)\ge|\text{Rank}(A)-\text{Rank}(B)|$

• $\text{Rank}(aa^\top)=1\quad$ (assuming $a\ne 0$ of course)