Subadditivity of matrix rank ($\mathrm{rank}(A+B)\le \mathrm{rank}A+\mathrm{rank}B$)

The image of the linear map $A+B$ is contained in the sum of the images of $A$ and $B$. Since the dimension of a vector space of this sum is at most the sum of their dimensions, matrix rank is subadditive. Iteration shows that a sum of $r$ matrices of matrix rank at most one has matrix rank at most $r$, over any field.