For n = 2, we have

$\displaystyle C_2=\left|\begin{array}{cc}\frac{1}{a_1+b_1}&\frac {1}{a_1+b_2} \\ \frac{1}{a_2+b_1}&\frac{1}{a_2+b_2}\end{array}\rig ht|$

$\displaystyle =\frac{1}{(a_1+b_1)(a_2+b_2)}-\frac{1}{(a_1+b_2)(a_2+b_1)}$

$\displaystyle =\frac{(a_1+b_2)(a_2+b_1)-(a_1+b_1)(a_2+b_2)}{(a_1+b_1)(a_2+b_2)(a_1+b_2)(a_ 2+b_1)}$

$\displaystyle =\frac{a_2b_2-a_2b_1-a_1b_2+a_1b_1}{(a_1+b_1)(a_2+b_2)(a_1+b_2)(a_2+b_1 )}$

$\displaystyle =\frac{(a_2-a_1)(b_2-b_1)}{(a_1+b_1)(a_2+b_2)(a_1+b_2)(a_2+b_1)}$