Let A be a real invertible $\displaystyle n$ x $\displaystyle n$ matrix. Show that

<x,y> ≡ $\displaystyle y^T$$\displaystyle A^T$$\displaystyle Ax$ = $\displaystyle (Ay)^T$$\displaystyle (Ax)$

defines an inner product in ℝ^n, where x and y are column vectors in ℝ^n. What happens when A is not invertible? (Note : $\displaystyle M^T$ is the transpose of matrix M, obtained by interchaging the rows and colums of M).