Rank of a product of matrices

Statement

Let $F$ be a field, let $A$ be an $m \times n$ matrix and $B$ an $n \times p$ matrix over $F$. Then $$\rank(AB) \le \min(\rank A, \rank B),$$ where $\rank$ is the rank of a matrix.

Remarks

The column space of $AB$ is contained in the column space of $A$, so $\rank(AB) \le \rank A$. The row space of $AB$ is contained in the row space of $B$, so $\rank(AB) \le \rank B$. If $A$ is invertible, then $\rank(AB) = \rank B$; if $B$ is invertible, then $\rank(AB) = \rank A$. A lower bound is Sylvester's rank inequality: $\rank A + \rank B - n \le \rank(AB)$.

Explanations

L1 human

Multiplying by a matrix can never create new independent directions. So the rank of a product is at most the rank of each factor. Multiplying by an invertible matrix keeps the rank.