Prove the following identities – |(2y,y-z-x,2y)(2z,2z,z-x-y)(x-y-z,2x,2x)| = (x + y + z)^3 ​

Question

Prove the following identities –

\(\begin{vmatrix} 2y & y-z-x & 2y \\[0.3em] 2z & 2z & z-x-y \\[0.3em] x-y-z &2x &2x \end{vmatrix}\) = (x + y + z)³

Answer 1

Let Δ = \(\begin{vmatrix} 2y & y-z-x & 2y \\[0.3em] 2z & 2z & z-x-y \\[0.3em] x-y-z &2x &2x \end{vmatrix}\) 

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 R₁→ R₁ + R₂, we get 

Expanding the determinant along R₁, we have 

Δ = (x + y + z)(1)[0 – (–(x + y + z)(x + y + z))] 

⇒ Δ = (x + y + z)(x + y + z)(x + y + z) 

∴ Δ = (x + y + z)³

Thus,

\(\begin{vmatrix} 2y & y-z-x & 2y \\[0.3em] 2z & 2z & z-x-y \\[0.3em] x-y-z &2x &2x \end{vmatrix}\) = (x + y +z)³