Let z1, z2, z3 ∈ C and f(z1,z2,z3)=sgn111z1z2z3z¯1z¯2z¯3 where sgn x=x|x| if x≠0 and sgn(0)=0, then for a≠0,

111az1az2az3a¯z¯1a¯z¯2a¯3z¯3=|a|2111z1z2z3z¯1z¯2z¯3 As |a|2 > 0 we get f(z1, z2, z3)=f(az1, az2, az3)

Let z1, z2, z3 ∈ C and f(z1,z2,z3)=sgn111z1z2z3z¯1z¯2z¯3 where sgn x=x|x| if x≠0 and sgn(0)=0, then for a≠0,

111az1az2az3a¯z¯1a¯z¯2a¯3z¯3=|a|2111z1z2z3z¯1z¯2z¯3 As |a|2 > 0 we get f(z1, z2, z3)=f(az1, az2, az3)

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Let $z_{1}, z_{2}, z_{3} \in C$ and $f (z_{1}, z_{2}, z_{3}) = sgn |\begin{matrix} 1 & 1 & 1 \\ z_{1} & z_{2} & z_{3} \\ {\bar{z}}_{1} & {\bar{z}}_{2} & {\bar{z}}_{3} \end{matrix}|$ where sgn $x = \frac{x}{| x |}$ if $x \neq 0$ and $s g n (0) = 0,$ then for $a \neq 0,$

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$f (z_{1}, z_{2}, z_{3}) = f (a z_{1}, a z_{2} a z_{3})$

$f (z_{1}, z_{2}, z_{3}) = f (a z_{3}, a z_{2}, a z_{1})$

none of these

$f (z_{1}, z_{2}, z_{3}) = f (a z_{2}, a z_{1}, a z_{3})$

answer is A.

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Detailed Solution

$|\begin{matrix} 1 & 1 & 1 \\ a z_{1} & a z_{2} & a z_{3} \\ \bar{a} {\bar{z}}_{1} & \bar{a} {\bar{z}}_{2} & {\bar{a}}_{3} {\bar{z}}_{3} \end{matrix}| = | a |^{2} |\begin{matrix} 1 & 1 & 1 \\ z_{1} & z_{2} & z_{3} \\ {\bar{z}}_{1} & {\bar{z}}_{2} & {\bar{z}}_{3} \end{matrix}|$