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

Question

Prove the following identities –

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

Answer 1

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

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 C₁, we have 

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

⇒ Δ = (x + y + z)(x – z)(x – z) 

∴ Δ = (x + y + z)(x – z)²

Thus,

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