Prove the following identities : |((b+c)^2,b^2,bc)((c+a)^2,b^2,ca)((a+b)^2,c^2,ab)|=(a-b) = (a - b) (b - c) (c - a) (a + b+ c)(a2 + b2 + c2) ​

Question

Prove the following identities :

\(\begin{vmatrix} (b+c)^2 & b^2 & bc \\[0.3em] (c+a)^2 & b^2 &ca \\[0.3em] (a+b)^2 &c^2 & ab \end{vmatrix}\) 

= (a - b) (b - c) (c - a) (a + b+ c)(a² + b² + c²)

Answer 1

L.H.S = \(\begin{vmatrix} (b+c)^2 & b^2 & bc \\[0.3em] (c+a)^2 & b^2 &ca \\[0.3em] (a+b)^2 &c^2 & ab \end{vmatrix}\) 

Applying, 

C₁→C₁ + C₂ – 2C₃

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

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

= R.H.S 

Hence, proved.