linear projection $x_{W}=A(A^TA)^{-1}A^Tx$, isn't projector eliminate itself?

Question

I am wondering isn't this projector itself is identity matrix?

$$x_{W}=A(A^TA)^{-1}A^Tx\\projector=A(A^TA)^{-1}A^T=AA^{-1}(A^T)^{-1}A^T=I$$

Thank you for any comment