Question

Evaluate: `|(3x, -x + y, -x + z),(x - y, 3y, z - y),(x - z, y - z, 3z)|`

Answer 1

We have, `|(3x, -x + y, -x + z),(x - y, 3y, z - y),(x - z, y - z, 3z)|`

[Applying C₁ → C₁ + C₂ + C₃]

= `|(x + y + z, -x + y, -x + z),(x + y + z, 3y, z - y),(x + y + z, y - z, 3z)|`

[Taking (x + y + z) common from colmn C₁]

= `(x + y + z)|(1, -x + y, -x + z),(1, 3y, z - y),(1, y - z, 3z)|`

[Applying R₁ → R₂ – R₁ and R₃ → R₃ – R₁]

= `(x + y + z)|(1, -x + y, -x + z),(0, 2y + x, x - y),(0, x - z, 2z + x)|`

[Applying C₂ → C₂ – C₃]

= `(x + y + z)|(1, -x + y, -x + z),(0, 3y, x - y),(0, -3z, 2z + x)|`

[Expanding along first column]

= `(x + y + z) * 1[3y(2z + x) + (3z)(x - y)]`

= (x + y + z)(3yz + 3yx + 3xz)

= 3(x + y + z)(xy + yz + zx)