Suppose $P$ is the projection matrix onto the column space of $A$. Then $$ P = A(A^T A)^{-1}A^T \quad\text{and}\quad PA=A $$ However, taking determinants of this equation gives $$ \det(PA)=\det(P) \det(A) = \det(A) $$
Assuming A is invertible this seems to imply $\det(P)=1$, although I know it should be 0. Can anyone point out my mistake please?