Cauchy-Schwarz inequality of determinants:

for $A_{n\times k}$, $B_{n\times k}$, and $B'B$ non-singular, we have

$|A'B|^2\leq |A'A||B'B|$

I was wondering what's the sufficient and necessary conditions for the equality to hold.

I know a sufficient condition:

when $\exists C_{k\times k}$, such that $A=BC$, then the equality holds.

Is this also a necessary condition?

Thanks.