Show that $P$ is symmetric.

Question

Let $P$ = $A(A^TA)^{-1}A^T$, where A is an m x n matrix with rank $n$.

I feel like this is wrong, but here is my attempt: $A(A^TA)^{-1}A^T$ = $AA^{-1}(A^T)^{-1}A^T$ = $I$ And $I^T$ = $I$, so the matrix is symmetric.

Answer 1

Your approach is only correct if $n=m$. If $n\not=m$, then $A^{-1}$ does not exist. $P$ is symmetric if $P^T=P$, so that's what you have to show.

$$P^T=\left(A\left(A^TA\right)^{-1}A^T\right)^T=\left(A^T\right)^T \left(\left(A^TA\right)^{-1}\right)^TA^T$$ $$=A\left(\left(A^TA\right)^{T}\right)^{-1}A^T=A\left(A^T\left(A^T\right)^T\right)^{-1}A^T=A\left(A^TA\right)^{-1}A^T=P$$