First slide

Introduction to Determinants

Question

For the equation $|\begin{array}{ccc} 1 & x & x^{2} \\ x^{2} & 1 & x \\ x & x^{2} & 1 \end{array}| = 0$ ,

Moderate

A

There are exactly two distinct roots
B

There is one pair of equation real roots
C

There are three pairs of equal roots
D

Modulus of each root is 2

Solution

$R_{1} \to R_{1} + R_{2} + R_{3} and take 1 + x + x^{2} as common Δ = (1 + x + x^{2}) |\begin{array}{ccc} 1 & 1 & 1 \\ x^{2} & 1 & x \\ x & x^{2} & 1 \end{array}| = {(1 - x^{3})}^{2}$
Therefore, $Δ = 0$ , has roots $1, 1, ω, ω, ω^{2}, ω^{2}$ .

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