Is the matrix invertible?

Question

True/False:

If $A$ is a $m×r$ matrix with $r$ independent columns and $B$ is a $r ×n$ matrix with $r$ independent rows, then $AB$ is invertible.

I reasoned like this: Since the resultant matrix is not square, $AB$ is not invertible. Is there a better way to prove this?

Answer 1

You are correct if $n\ne m$.

If $n=m$, then:

• If $r<n$ then it can be shown that $AB$ will have a rank of $r$ (can you show this?). Since $r<n$, $AB$ is not invertible in that case.

• If $r=n$ then $AB$ is invertible.

• The case $r>n$ is impossible (why?).