Question

Prove that: `|(y + z, z, y),(z, z + x, x),(y, x, x + y)|` = 4xyz

Answer 1

`|(y + z, z, y),(z, z + x, x),(y, x, x + y)|` 

[Applying C₁ → C₁ + C₂ + C₃]

= `|(2(y + z), z, y),(2(z + x), z + x, x),(2(y + x), x, x + y)|`

= `2|(y + z, z, y),(z + x, z + x, x),(x + y, x, x + y)|`

[Applying C₁ → C₁ – C₂]

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

[Applying C₃ → C₃ – C₁]

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

[Applying R₃ → R₃ – R₁]

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

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

= `2y[2xz]`

= 4xyz