Question

The value of x obtained from the equation `|(x + a, b, c),(c, x + b, a),(a, b, x + c)|` = 0 will be ______.

Options

• 0 and – (a + b + c)

• 0 and (a + b + c)

• 1 and (a – b – c)

• 0 and (a² + b² + c²)

Answer 1

The value of x obtained from the equation `|(x + a, b, c),(c, x + b, a),(a, b, x + c)|` = 0 will be 0 and – (a + b + c).

Explanation:

We have, `|(x + a, b, c),(c, x + b, a),(a, b, x + c)|`

Applying C₁ `rightarrow` C₁ + C₂ + C₃,

`|(x + a + b + c, b, c),(x + a + b + c, x + b, a),(x + a + b + c, b, x + c)|` = 0

= `(x + a + b + c) |(1, b, c),(1, x + b, a),(1, b, x + c)|` = 0

Applying R₂ `rightarrow` R₂ – R₁ and R₃ `rightarrow` R₃ – R₁,

`(x + a + b + c) |(1, b, c),(0, x, a - c),(0, 0, x)|` = 0

`\implies` (x + a + b + c) (x² – 0) = 0

`\implies` x = 0 or x = – (a + b + c)