Question

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

Answer 1

! a-b b-c c-a !
! b-c c-a a-b ! = 0
! c-a a-b b-a !

Answer 2

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

using the properties of Determinant

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

exchange R1 <=> R3 and R2 <=> R3 in second Determinant

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

now we can see that both the determinants are same,so on subtraction result will be zero

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