Question

If a, b, c are non-zeros, then the system of equations (α + a)x + αy + αz = 0, αx + (α + b)y + αz = 0, αx+ αy + (α + c)z = 0 has a non-trivial solution if

Options

• α–1 = – (a^–1 + b^–1 + c^–1)

• α^–1 = a + b + c

• α + a + b + c = 1

• None of these

Answer 1

α^–1 = – (a^–1 + b^–1 + c^–1)

Explanation:

Given system of equation AX = O

Where A = `[(alpha + a, alpha, alpha),(alpha, alpha + b, alpha),(alpha, alpha, alpha + c)] x X = [(x),(y),(z)], O = [(0),(0),(0)]`

For non-trival solution,

D = 0

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

`R_1 -> R_1 - R_2, R_2 -> R_2  - R_3`

⇒ `|(a, -b, 0),(0, b, -c),(alpha, alpha, alpha + c)|` = 0

⇒ abα + abc + acα + bcα = 0

⇒ abc = – α(ab + bc + ca)

⇒ `1/alpha = -(1/c + 1/a + 1/b)`

α^–1 = – (a^–1 + b^–1 + c^–1)