My task was to show that certain matrices are idempotent, that is, ${AA} = {A}$. I struggled with the proof for one case and when I look at the solution, I have problems understanding one step.
Prove that the matrix $I_n - A (A^TA)^{(-1)}A^T $ is idempotent:
\begin{equation} \begin{split} & I_n - A(A^T A)^{-1} A^T) \times (I_n - A(A^T A)^{-1} A^T) \\ &= I_n -2A(A^TA)^{-1} A^T + A(A^TA)^{-1} A^TA(A^TA)^{-1} A^T \\ & = I_n -2A(A^TA)^{-1} A^T + A(A^TA)^{-1} A^T \\ & = I_n - A (A^TA)^{-1}A^T \end{split} \end{equation}
I am struggling to follow why $A(A^TA)^{-1} A^TA(A^TA)^{-1} A^T \Rightarrow A(A^TA)^{-1} A^T$ in step 2 to 3.
By associativity, notice that: \begin{align*} A(A^TA)^{-1} A^TA(A^TA)^{-1} A^T &= A(A^TA)^{-1} \left[(A^TA)(A^TA)^{-1}\right] A^T \\ &= A(A^TA)^{-1} \left[I_n\right] A^T \\ &= A(A^TA)^{-1} A^T \\ \end{align*}