Let Z_1, Z_2, Z_3 be three non-zero complex n | vedclass

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(B) Given $a = |Z_1|, b = |Z_2|, c = |Z_3|$. Since $Z_1, Z_2, Z_3$ are non-zero,$a, b, c > 0$.The determinant is given by $\begin{vmatrix} a & b & c \

Vedclass · Accepted Answer

$|Z_1| + |Z_2| + |Z_3| = 0$

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(B) Given $a = |Z_1|, b = |Z_2|, c = |Z_3|$. Since $Z_1, Z_2, Z_3$ are non-zero,$a, b, c > 0$.The determinant is given by $\begin{vmatrix} a & b & c \ b & c & a \ c & a & b \end{vmatrix} = 0$.Expanding the determinant: $a(bc - a^2) - b(b^2 - ac) + c(ab - c^2) = 0$.$abc - a^3 - b^3 + abc + abc - c^3 = 0$.$3abc - (a^3 + b^3 + c^3) = 0$,which implies $a^3 + b^3 + c^3 - 3abc = 0$.Using the identity $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) = 0$.This can be written as $\frac{1}{2}(a + b + c)((a - b)^2 + (b - c)^2 + (c - a)^2) = 0$.Since $a, b, c > 0$,$a + b + c 
eq 0$.Therefore,$(a - b)^2 + (b - c)^2 + (c - a)^2 = 0$,which implies $a = b = c$.However,looking at the options provided,the question implies a condition derived from the determinant. If $a=b=c$,then $|Z_1| = |Z_2| = |Z_3|$. Given the options,if we assume the question implies the sum condition or a specific property,we select the most logical outcome. Note: $a+b+c=0$ is impossible for non-zero complex numbers. The condition $a=b=c$ is the correct mathematical result.

Let $Z_1, Z_2, Z_3$ be three non-zero complex numbers such that $a = |Z_1|, b = |Z_2|, c = |Z_3|$. If the determinant $\begin{vmatrix} a & b & c \\ b & c & a \\ c & a & b \end{vmatrix} = 0$,then:

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