How could I know the first two second derivative (4,-1) should be placed as row or column vector when they are placed together in the Hessian matrix?
Can I conclude that since the inner two second derivative is always the same, it doesn’t matter whether (4,-1) is placed as row or column vector?
Yes, Hessian matrices are symmetric as long as the second derivatives are continuous, so in the cases we will see H = H^T.