TITLE: Sum of two rank deficient matrices
QUESTION [5 upvotes]: Suppose I have two $m\times n$, where $m>n$, matrices $A$ and $B$. The rank of $A$ and the rank of $B$ are strictly less than $n$.  
Are there any (general) sufficient conditions under which one can guarantee that the rank of sum $A+B$  is strictly less than $n$?

REPLY [6 votes]: Noting that 

$\qquad\text{im}(A+B)\;\text{is a subspace of im}(A) + \text{im}(B)$,

an easy sufficient condition is
 
$\qquad\text{rank}(A)+\text{rank}(B) < n$, 

since then
\begin{align*}
\text{rank}(A+B) 
&=\dim(\text{im}(A+B))\\[4pt]
&\le\dim(\text{im}(A) + \text{im}(B))\\[4pt]
&\le\dim(\text{im}(A)) + \dim(\text{im}(B))\\[4pt]
&=\text{rank}(A) + \text{rank}(B)\\[4pt]
&<n
\end{align*}