I am trying to calculate the extrema of the function $$f(x,y) = \sin(x)\sin(y)\sin(x+y)$$ with the constraint $0 < x, y,x+y < \pi$. I have determined the critical point to be $\left( \frac{\pi}{3} , \frac{\pi}{3} \right)$. Now I want to determine whether this is a maximum or a minimum. For that, I would obviously need the Hessian matrix and its determinant or eigenvalues. I get
$$H(f)\left( \frac{\pi}{3} , \frac{\pi}{3} \right) = \begin{pmatrix} -\sqrt{3} & -\frac{\sqrt{3}}{2} \\ -\frac{\sqrt{3}}{2} & -\sqrt{3} \\ \end{pmatrix}$$
Its determinant is easily calculated to be $\det(H) = 2.25$. However, its eigenvalues are both negative and judging from a plot the point should actually refer to a maximum. Something's got to be wrong here, since the determinant should be negative.
I'll restrict the discussion to symmetric marices, since we are talking about the Hessian matrix, which is symmetric.An equivalent condition for negative definiteness, but much easier to compute (especially if the matrix is large)is that the principal minors of odd order must be negative and the principal minors of even order must be positive. Even easier, and also equivalent, is that, for an nxn matrix, the determinant of the upper left rxr corner must be negative if r is odd and positive if r is even for $1\le r \le n$. Buy and study Ferrar's old book "Algebra-Atext-book of determinants, matrices and algebraic forms" (Oxford U.P. 1957)