Question

Using properties of determinants, prove the following:

|(3x, -x + y, - x + z), (x - y, 3y, z - y), (x - z, y -z, 3z)| = 3(x + y + z)(xy + yz + zx)

Answer 1

Applying C₁ → C₁ + C₂ + C₃

Taking out (x + y + z) along C₁, we get

Applying R₂ → R₂ – R₁; R₃ → R₃ – R₁

Applying C₂ → C₂ – C₃

Expanding along I column, we get 

D = (x + y + z)[(3y (x + 2z) + 3z (x – y)] 

= 3(x + y + z)[xy + 2z + 2yz + xz – yz] 

= 3(x + y + z)(xy + yz + zx) = R.H.S.