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

Question

Prove the following identities –

\(\begin{vmatrix} a+x & y & z \\[0.3em] x & a+y & z\\[0.3em] x & y & a+z \end{vmatrix}\) = a²( a + x + y + z)

Answer 1

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

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 C₁→ C₁ + C₂, we get

Expanding the determinant along C₁, we have 

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

⇒ Δ = (a + x + y + z)(a)(a) 

∴ Δ = a²(a + x + y + z)

Thus,

\(\begin{vmatrix} a+x & y & z \\[0.3em] x & a+y & z\\[0.3em] x & y & a+z \end{vmatrix}\)= a²(a + x + y + z)