Question

Prove that,

\(\begin{vmatrix} b + c & a + b & a \\[0.3em] c + a & b + c & b \\[0.3em] a + b &c + a &c \end{vmatrix}\) = a³ + b³ + c³ - 3abc

Answer 1

LHS = \(\begin{vmatrix} b + c & a + b & a \\[0.3em] c + a & b + c & b \\[0.3em] a + b &c + a &c \end{vmatrix}\) 

= (a + b + c)[0 – 0 + 1{(a – c)(b – c) – (a – b) (b – a)}]

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

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

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

= a³ + b³ + c³ – 3abc – R.H.S. Proved