how does the determinant of a matrix factor in a matrix product determines the linear dependence of the columns?

Question

A certain theorem states that the matrices $N$ (3 by 2), $P$ (3 by 2) and $W$ (2 by 2) are related by the equation $$N=-PW$$ Then a corollary states that the columns of $N$ are linearly dependent iff the determinant of $W$ is null, seeing as the columns of $P$ are linearly independent. Why is that? I would of course understand this in the context of the product of square matrices, but here I am a little confused.

Answer 1

A linear relation between columns of $N$ is of the form $NX=0$ for some $2\times1$ matrix $X\ne0$. But $NX=0$ is equivalent to $P(-WX)=0$, where $-WX\ne0$ is $2\times1$, which implies $P$'s columns are linearly dependent. The converse is also true if $W$ is invertible: if $PY=0$ where $Y\ne0$ is $2\times1$, then $N(-W^{-1}Y)=0$ where $-W^{-1}Y\ne0$ is also $2\times1$.