Calculate the distance between x and y using inner product

Question

I am having troubles with this question. Let $x = (2,-1,3)$ and $y = (5,-2,2)$ in $R^3$

Calculate the distance between x and y using the inner product with matrix

$ A = \begin{bmatrix}-2&&0&&1\\1&&-1&&2\\3&&-1&&-1\end{bmatrix}$

Answer 1

The distance is $d(x,y)=\sqrt{(x-y)^+A(x-y)}$. An issue however is that an inner product matrix should usually be symmetric & positive, which is not the case here ($A^+\neq A$, and $(x-y)^+A(x-y)=-34<0$.