Sum of two rank deficient matrices

Question

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 $n$?

Answer 1

If $\ker(A)\cap \ker(B)\neq \{0\}$, then you know that $\ker(A+B)$ will not be empty. This is because $(A+B)x=Ax+Bx=0$ if $Ax=Bx=0$.

Of course this is not a neccesary condition, but it is sufficient.

Answer 2

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*}