prove that: determinant | a b c a^2 b^2 c^2 bc ca ab|= (a-b)(b-c)(c-a)(ab+bc+ca) - Maths - Determinants

Question

prove that: determinant | a b c a^2 b^2 c^2 bc ca ab|=
(a-b)(b-c)(c-a)(ab+bc+ca)

Anuradha Sharma · Accepted Answer

Given determinant is, A=abca2b2c2bccaabNow performing C1→C1-C2 and C2→C2-C3=a-bb-cca2-b2b2-c2c2bc-caca-abab=a-bb-cca-ba+bb-cb+cc2-ca-b-ab-cab=a-b1b-cca+bb-cb+cc2-c-ab-cab=a-bb-c11ca+bb+cc2-c-aabAgain performing  C1→C1-C2, =a-bb-c01ca-cb+cc2a-c-aab=a-bb-ca-c01c1b+cc21-aabNow performing R2→R2-R3=a-bb-ca-c01c0b+c+ac2-ab1-aabNow expanding the determinant along C1, we get, =a-bb-ca-c0-0+1c2-ab-bc-c2-ac=a-bb-cc-aab+bc+cahence proved