If $P$ is a $7\times7$ matrix of rank $4$ and if $\hat{a}$ be any vector in $\mathbb{R}^7$, then to find minimum rank of $S=P+\hat{a}\hat{a}^t$ .

Question

Actually, I have been trying over this problem during finding out a tight lower bound for the rank of the matrix $(A+B)$ where $A$ and $B$ are $m\times n$ matrices .

Though we know an upper bound that is $r(A)+r(B)- \max\{ rank A, rank B\}$ but I can't figure out a lower bound .

Obviously, the rank of $\hat{a}\hat{a}^t$ is $1$ in this problem, and if $\hat{a}$ lies outside the column space of $P$, then the rank of $S$ is at least $4$ . Help me out what happens if $\hat{a}$ lies in $C(P)$ . But, the answer provided is $3$ .

Answer 1

Hint. Write $P=S+(-\hat{a}\hat{a}^t)$ and consider the rank of $P$.