This is problem I heard briefly, not even sure if it's well-stated, but here we are:
Prove that if $x_i>0, y_j>0$ then $\det e^{x_iy_j}\ge 0$. I think there was also a hint to use "generalized Vandermonde determinant", but I can't figure out even what this generalized determinant is.
May be you have some ideas? thanks in advance!
This property is erroneous : take the case of a $2 \times 2$ matrix with :
$$x_1=3, x_2=2 , y_1=1, y_2=2$$
The determinant of the matrix is
$$e^{x_1y_1}e^{x_2y_2}-e^{x_1y_2}e^{x_2y_1}$$
$$=e^{3}e^{4}-e^{6}e^{2}=e^{7}-e^{8} \color{red}{< 0}$$