`[[x,x^2, 1+x^2],[y, y^2, 1+y^2],[z, z^2, 1+z^2]]` where x,y,z are distinct. What is |A|.

Options

A. 0
B. `x^(2)y-y^(2)x+xyz`
C. `(x-y)(y-z)(z-x)`
D. xyz

Answer

C

Explanation

`A=[{:(x,x^(2),1+x^(2)),(y,y^(2),1+y^(2)),(z,z^(2),1+z^(2)):}]`
`|A|=|{:(x,x^(2),1+x^(2)),(y,y^(2),1+y^(2)),(z,z^(2),1+z^(2)):}|`
Applying `R_(1) rarr R_(1) - R_(2) and R_(2) rarr R_(2) - R_(3)`
`|A|=|{:(x-y,(x-y)(x+y),(x-y)(x+y)),(y-z,(y-z)(y+z),(y-z)(y+z)),(" "z," "z^(2)," "1+z^(2)):}|`
`=(x-y)(y-z)|{:(1,x+y,x+y),(1,y+z,y+z),(1," "z^(2),1+z^(2)):}|`
Applying `C_(3) rarr C_(3) - C_(2)`
`|A|=(x-y)(y-z)|{:(1,x+y,0),(1,y+z,0),(z," "z^(2),1):}|`
`=(x-y)(y-z)[1{y+z-(x+y)]`
`=(x-y)(y-z)(z-x)`