सिद्ध कीजिए कि |(1 1 1),(a b c),(a^3 b^3 c^3)| = (b-c)(c-a)(a-b)(a+b+c)

Question

सिद्ध कीजिए कि \(\begin{vmatrix}1&1&1\\ a&b&c\\ a^3&b^3&c^3\end{vmatrix}\) = (b-c)(c-a)(a-b)(a+b+c)

Answer 1

L.H.S. =  \(\begin{vmatrix}1&1&1\\ a&b&c\\ a^3&b^3&c^3\end{vmatrix}\) 

R1 के सापेक्ष प्रसार करने पर,

= (a – b) (b – c) [(b² + c²+ bc) – (a² + b² + ab)]

= (a – b) (b – c) (b² + c²+ bc – a² – b²– ab)

= (a – b) (b – c) [bc + c² – a² – ab]

= (a – b) (b – c) [bc – ab + c² – a²]

= (a – b) (b – c) [b(c – a) + (c² – a²)]

= (a – b) (b – c) (c – a) (b + c + a)

= R.H.S.
इति सिद्धम्।