prove the positivity of n by n matrix determinant

Question

I am trying to show that [det $(\frac{1}{\eta_{j}-\lambda_{i}})_{ij}$] is positive, such that $\lambda_{1}, \ldots , \lambda_{n}$ and $\eta_{1}, \ldots , \eta_{n}$ are real numbers which satisfy the inequalities $$ \lambda_{1} < \eta_{1} \leq \lambda_{2} < \eta_{2} \leq \ldots \leq \lambda_{n} < \eta_{n} .$$