Question

Evaluate: `|(1,a,a^2 - bc),(1,b,b^2 - ca),(1,c,c^2 - ab)|`

Answer 1

`|(1,a,a^2 - bc),(1,b,b^2 - ca),(1,c,c^2 - ab)|`

= `|(1,a,a^2),(1,b,b^2),(1,c,c^2)| + |(1,a,- bc),(1,b,- ca),(1,c,- ab)|`  ...(By property 6)

= `|(1,a,a^2),(1,b,b^2),(1,c,c^2)| - |(1,a,bc),(1,b,ca),(1,c,ab)|`  ...(Take out -1 from C₃)

= `|(1,a,a^2),(1,b,b^2),(1,c,c^2)| - 1/"abc" |(a,a^2,abc),(b,b^2,abc),(c,c^2,abc)|`  .....(Multiply R₁ by a, R₂ by b, R₃ by c and divide the determinant by abc)

= `|(1,a,a^2),(1,b,b^2),(1,c,c^2)| - "abc"/"abc" |(a,a^2,1),(b,b^2,1),(c,c^2,1)|`  ....(Take out abc from C₃)

= `|(1,a,a^2),(1,b,b^2),(1,c,c^2)| + |(a,1,a^2),(b,1,b^2),(c,1,c^2)|`   ..... (C₂ ↔ C₃)

= `|(1,a,a^2),(1,b,b^2),(1,c,c^2)| - |(1,a,a^2),(1,b,b^2),(1,c,c^2)|`   ..... (C₁ ↔ C₂)

= 0