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Let $A$ be a $7\times 7$ matrix such that it has rank $3$ and $a$ be $7\times 1$ column vector. Then least possible rank of $A+(a a^T)$ is? ($a^T$ is transpose of the vector)

Intuitively I think it's $2$ as $a a^T$ has rank $1$, and the most it could affect is one row of $A$.

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    For an introduction to posting with [math notation](http://math.stackexchange.com/help/notation), see that portion of the site FAQ and its links to other resources about `MathJax` and $\LaTeX$. – 2017-02-15
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    While not literally the case that $aa^T$ can only affect one row of $A$, it is a good special case to motivate your intuition. **Hint:** Consider left multiplying $A+aa^T$ by an invertible matrix which reduces the general case to the single row affected case of your intuition. – 2017-02-15

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your intuition is right. instead of working with rows as suggested by hardmath, i will work with the columns of $A.$ without loss of generality, we can assume that the first three columns $a_1, a_2$ and $a_3$ are linearly independent.the column space of $Aa + aa^\top$ is $\{Ax + aa^\top x = Ax + (a^\top x)\, a\ \}$ the dimension space is two iff there is an $x$ such that $a_1x_j + (a^\top x) a = 0,\ j = 1, 2, 3.$ in other cases the dimension is either three or four.