Is $\operatorname{null}(B) \subset \operatorname{null}(A)$?

$A$ and $B$ are two symmetric and PSD matrices. Also,

$B = A + ee^T$.

How can one prove that $\operatorname{null}(B) \subset \operatorname{null}(A)$?

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Suppose $x \in \operatorname{null}(B)$, i.e., $Bx = 0$. What can you prove about $Ax$? –  Jonathan Christensen Dec 6 '12 at 21:54
We somehow have to prove that x satisfies Ax = 0, and thereby prove that x belongs to null(A) too? –  Ashwin Dec 6 '12 at 22:09

Hint: look at $x^tBx=x^tAx+x^tee^tx$, where $x$ is such that $Bx=0$.