# Evaluate the following Determinant

Find $$\Delta=\begin{vmatrix} \frac{1}{a+x} &\frac{1}{b+x} &\frac{1}{c+x} \\ \frac{1}{a+y} &\frac{1}{b+y} &\frac{1}{c+y} \\ \frac{1}{a+z} &\frac{1}{b+z} &\frac{1}{c+z} \end{vmatrix}$$

I applied $C_1 \to C_1-C_2$ and $C_2 \to C_2-C_3$ we get

$$\Delta=\begin{vmatrix} \frac{b-a}{(a+x)(b+x)} &\frac{c-b}{(b+x)(c+x)} &\frac{1}{c+x} \\ \frac{b-a}{(a+y)(b+y)} &\frac{c-b}{(b+y)(c+y)}&\frac{1}{c+y} \\ \frac{b-a}{(a+z)(b+z)} &\frac{c-b}{(b+z)(c+z)} &\frac{1}{c+z} \end{vmatrix}$$