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we know that for all $A,B\in M_n(\mathbb{C})$ : $$ rank (A+B)\leq rank(A)+rank (B) $$ see here for a simple proof, but for which condition on the coefficients of $A$ and $B$ we can obtain a perfect equality.

more simply if we assume that $rank(A)=1$ what will be the condition on $B$ to have : $$ rank(A+B)=rank(B)+1 $$

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    $\begingroup$ It is necessary, but not sufficient, that the augmented matrix $[A \; B]$ have rank $rank(A) + rank(B)$. $\endgroup$ – Ben Grossmann Jun 18 '16 at 20:32
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I have found the answer here WHEN DOES $RANK(A+B) = RANK(A) + RANK(B)?$ the answer is that : $$ R(A)\cap R(B)=0 \\ R(A^*)\cap R(B^*)=0 $$

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