First slide

Introduction to Determinants

Question

If $|\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ bc & ca & ab \end{array}| = |\begin{array}{ccc} 1 & 1 & 1 \\ a & b & c \\ a^{3} & b^{3} & c^{3} \end{array}|$ , where a, b, c are distinct positive reals, then the possible values of abc is/are

Moderate

A

$\frac{1}{81}$
B

$\frac{1}{63}$
C

$\frac{1}{27}$
D

$\frac{1}{18}$

Solution

We have
$(a - b) (b - c) (c - a) = (a - b) (b - c) (c - a) (a + b + c) (a \neq b \neq c)$
$\Rightarrow a + b + c = 1$
As a, b and c are positive.
Using AM > GM, we get
$\begin{array}{l} \frac{a + b + c}{3} > (abc)^{\frac{1}{3}} \\ ∴ abc < \frac{1}{27} \end{array}$

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