# Find the determinant of a matrix with entries $\frac1{a_i+b_j}$

Find the determinant

$$\begin{vmatrix} \dfrac1{a_1+b_1} & \dfrac1{a_1+b_2} & \ldots & \dfrac1{a_1+b_n} \\ \dfrac1{a_2+b_1} & \dfrac1{a_2+b_2} & \ldots & \dfrac1{a_2+b_n} \\ \vdots & \vdots & \ddots & \vdots \\ \dfrac1{a_n+b_1} & \dfrac1{a_n+b_2} & \ldots & \dfrac1{a_n+b_n} \end{vmatrix}$$

I know you have to multiply -1 to the first row and add it to every rest rows, but I'm unsure of what to do after.

• look up the cauchy/hilbert matrix – abel Jun 8 '15 at 3:57