$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{a - b}&{b - c}&{c - a}\\{b - c}&{c - a}&{a - b}\\{c - a}&{a - b}&{b - c}\end{array}} \right| = 0$

L.H.S. $=$ $\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{a - b}&{b - c}&{c - a}\\{b - c}&{c - a}&{a - b}\\{c - a}&{a - b}&{b - c}\end{array}} \right|$

Applying ${C_1} \to {C_1} + {C_2} + {C_3}$ ,

we get
$\left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}{a - b + b - c + c - a}&{b - c}&{c - a}\\{b - c + c - a + a - b}&{c - a}&{a - b}\\{c - a + a - b + b - c}&{a - b}&{b - c}\end{array}} \right|$

$= \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}0&{b - c}&{c - a}\\0&{c - a}&{a - b}\\0&{a - b}&{b - c}\end{array}} \right| = 0$ [as ${C_1} = 0$]