Question

If a, b, g are the roots of x³ + ax² + b = 0, then the

determinant D = $\left| \begin{matrix} \alpha & \beta & \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{matrix} \right|$equals:

• A. –a³
• B. a³ –3b
• C. a² –3b
• D. a³

Answer 1

Answer: (D)

a³

Answer 2

a + b + g = –a, $\sum_{}^{}{\alpha\beta = 0}$

using R₁ ® R₁ + R₂ + R₃

D= $\left| \begin{matrix} \alpha + \beta + \gamma & \alpha + \beta + \gamma & \alpha + \beta + \gamma \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{matrix} \right|$D = – a $\left| \begin{matrix} 1 & 1 & 1 \\ \beta & \gamma & \alpha \\ \gamma & \alpha & \beta \end{matrix} \right|$

Apply C₂ ® C₂ – C₁, C₃ ® C₃ – C₁

or D = – a $\left| \begin{matrix} 1 & 0 & 0 \\ \beta & \gamma - \beta & \alpha - \beta \\ \gamma & \alpha - \gamma & \beta - \gamma \end{matrix} \right|$