Question

If a+ b+ c not equal to zero , and | a b c, b c a, c a b| = 0, then using properties of determinants , prove that a= b= c ( urgently needed from experts)

Answer 1

Answer:

Foemula used:

$a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)$

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

Expanding along first row

$a(bc-a^2)-b(b^2-ca)+c(ab-c^2)=0\\\\abc-a^3-b^3+abc+abc-c^3=0\\\\a^3+b^3+c^3-3abc=0\\\\(a+b+c)(a^2+b^2+c^2-ab-bc-ca)=0\\\\But\:a+b+c\neq0\\\\a^2+b^2+c^2-ab-bc-ca=0\\\\a^2+b^2+c^2=ab+bc+ca$

This is possible only when

a=b=c