Prove the following identities – |(a^2+1,ab,ac)(ab,b^2+1,bc)(ca,cb,c^2+1)| = 1 + a^2 + b^2 + c^2 ​

Question

Prove the following identities – 

\(\begin{vmatrix} a^2+1 &ab & ac \\[0.3em] ab& b^2+1 & bc \\[0.3em] ca & cb & c^2+1 \end{vmatrix}\) = 1 + a² + b² + c²

Answer 1

Let Δ = \(\begin{vmatrix} a^2+1 &ab & ac \\[0.3em] ab& b^2+1 & bc \\[0.3em] ca & cb & c^2+1 \end{vmatrix}\) 

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. 

Applying C₂→ C₂ – C₁, we get

Expanding the determinant along R₁, we have

Δ = (1 + a² + b² + c²)[(1)(1) – (0)(0)] – 0 + 0 

⇒ Δ = (1 + a² + b² + c²)(1) 

∴ Δ = 1 + a² + b² + c²

Thus,

\(\begin{vmatrix} a^2+1 &ab & ac \\[0.3em] ab& b^2+1 & bc \\[0.3em] ca & cb & c^2+1 \end{vmatrix}\) = 1 + a² + b² + c²