Prove the following identities – |(b^2+c^2,ab,ac)(ba,c^2+a^2,bc)(ca,cb,a^2+b^2)| = 4a^2b^2c^2 ​

Question

Prove the following identities –

\(\begin{vmatrix} b^2+c^2 & ab & ac \\[0.3em] ba & c^2+a^2 & bc \\[0.3em] ca & cb & a^2+b^2 \end{vmatrix}\) = 4a²b²c²

Answer 1

Let Δ = \(\begin{vmatrix} b^2+c^2 & ab & ac \\[0.3em] ba & c^2+a^2 & bc \\[0.3em] ca & cb & a^2+b^2 \end{vmatrix}\) 

Recall that the value of a determinant remains same if we apply the operation R_i→ R_i + kR_j or C_i→ C_i + kC_j.

Expanding the determinant along R₁, we have 

Δ = 0 + (2c²)[(b²)(a² + b² – c²)] + (–2b²)[–(c²)(c² + a² – b²)] 

⇒ Δ = 2b²c²(a² + b² – c²) + 2b 2c 2(c² + a² – b²) 

⇒ Δ = 2b²c² [(a² + b² – c²) + (c² + a² – b²)] 

⇒ Δ = 2b²c²[2a²] 

∴ Δ = 4a²b²c

Thus,

\(\begin{vmatrix} b^2+c^2 & ab & ac \\[0.3em] ba & c^2+a^2 & bc \\[0.3em] ca & cb & a^2+b^2 \end{vmatrix}\) = 4a²b²c²