When does $AB$ have linearly independent columns, if $A$ and $B$ are non-square matrices?

Question

If

• $A$ is $m \times n$ ($m<n$), and its rows are independent

• $B$ is $n \times p$ ($p<n$), and its columns are independent

• We also know $m\ge n$.

does $AB$ have linearly independent columns?

Or what additional requirements are needed for $AB$ to have linearly independent columns?

Answer 1

Since $A$ has rank $m$ and $B$ has rank $p$, $AB$ has rank at most $\min(m,p)$. $AB$ is $m\times p$, so it could have linearly independent columns if $m \ge p$, but not if $m < p$.

Answer 2

For this we need $AB $ to have rank $p $ (since $p$ columns). All we are garanteed however is that the rank of $AB$ is less orequal to the minimum of $\{m,n,p\}$. So this will fail for example when $m <p$.