Using properties of determinants, prove that |(b+c)^2, a^2, bc), ((c+a)^2, b^2, ca), ((a+b)^2, c^2, ab)|

Question

Using properties of determinants, prove that

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

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

Answer 1

Expanding along C₁ we get

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

= (b - a) (c - a) (a² + b² + c²) [- b² - ab + c² + ac]

= (b - a) (c - a) (a² + b² + c²) [(c² - b²) + (ca - ab)]

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

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

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

Hence proved