Hat matrix: various results in linear algebra.

Question

In the question, the hat matrix $H = I-x(x^Tx)^{-1}x^T$ is an $n$ x $n$ matrix. $x$ is a $n$ x $p$ vector, whose columns are linearly dependent. How do I show that $Hx = 0$, $H^T = H$, and $H^2 = H$. I'm a bit confused as when I try and manipulate $H$, I get $x(x^Tx)^{-1}x^T = xx^{-1}(x^T)^{-1}x^T = I$ which would imply $H = 0$? This obviously doesn't make sense. Where am I going wrong here?