First slide

Algebra of complex numbers

Question

Suppose $a, b, c \in C,$ and $| a | = | b | = | c | = 1$ and $a b c = a + b + c$ then $b c + c a + a b$ is equal to

Moderate

A

0
B

-1
C

1
D

none of these

Solution

$| a |^{2} = | b |^{2} = | c |^{2} = 1$
$\Rightarrow a \bar{a} = b \bar{b} = c \bar{c} = 1$
Now, $a b c = a + b + c$
$\Rightarrow \bar{a} \bar{b} \bar{c} = \bar{a} + \bar{b} + \bar{c}$
$\begin{array}{l} \Rightarrow \frac{1}{a b c} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \\ \Rightarrow b c + c a + a b = 1 \end{array}$

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