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Question

Question: If \(\left| \begin{matrix} x + \alpha & \beta & \gamma \\ \alpha & x + \beta & \gamma \\ \alpha & \b...

If x+αβγαx+βγαβx+γ\left| \begin{matrix} x + \alpha & \beta & \gamma \\ \alpha & x + \beta & \gamma \\ \alpha & \beta & x + \gamma \end{matrix} \right| = 0, then x =

A

0, – (a + b + g)

B

0, a + b + g

C

1, a + b + g

D

0, a2 + b2 + g2

Answer

0, – (a + b + g)

Explanation

Solution

Let C1 + C2 + C3

Žx+α+β+γβγx+α+β+γx+βγx+α+β+γβx+γ\left| \begin{matrix} x + \alpha + \beta + \gamma & \beta & \gamma \\ x + \alpha + \beta + \gamma & x + \beta & \gamma \\ x + \alpha + \beta + \gamma & \beta & x + \gamma \end{matrix} \right| = 0 Ž (x + a + b + g)

1βγ1x+βγ1βx+γ\left| \begin{matrix} 1 & \beta & \gamma \\ 1 & x + \beta & \gamma \\ 1 & \beta & x + \gamma \end{matrix} \right| = 0 Let R3 ® R3 – R2; R2 ® R2 – R1

Ž (x + a + b + g) 1βγ0x00xx\left| \begin{matrix} 1 & \beta & \gamma \\ 0 & x & 0 \\ 0 & –x & x \end{matrix} \right| = 0 Ž (x + a + b + g) x2 = 0

\ x = 0, – (a + b + g)