\begin{bmatrix}\frac{2}{x_1^3x_2}&\frac{1}{x_1^2x_2^2}\\\frac{1}{x_1^2x_2^2}&\frac{2}{x_1 x_2^3}\end{bmatrix}
where $x_1$ and $x_2$ are positive real numbers.
Using Slyvester's Citerion:
$\frac{2}{x_2 x_1^3} > 0$
$\frac{2}{x_2 x_1^3}\frac{2}{x_1 x_2^3}-\frac{1}{x_1^2x_2^2}\frac{1}{x_1^2x_2^2} > 0$
The solution claims that this matrix is positive semi definite, but I don't see a way for the principle minors to be 0.
The determinant is $\frac3{x_1^4x_2^4}\neq 0$ so it can't have a $0$ eigenvalue. However it still is positive semi-definite (as well as positive definite)