Question

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

Answer 1

From L.H.S.

Any two rows or column of determinant are changed, then the sign of determinant also changed.

On applying C₁ → C₁ + C₂ + C₃

Take (a + b + c) common from C₁

On applying R₃ → R₃ - 2R₁

= – (a + b + c) [(b – c) (a + b – 2c) – (c – a) (c + a – 2b)]

= 3abc – a³ – b³ – c³

So, L.H.S = R.H.S,

Hence proved.