Question

Using the property of determinants and without expanding, find

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

Answer 1

Let $\Delta =\left|\begin{array}{ccc}a-b& b-c& c-a\\ b-c& c-a& a-b\\ c-a& a-b& b-c\end{array}\right|$
Applying $R_1=R_1+R_2+R_3$, we get,
$\Delta =\left|\begin{array}{ccc}0& 0& 0\\ b-c& c-a& a-b\\ c-a& a-b& b-c\end{array}\right|=0$
Since all the elements of first row are zero.