The system of equations α x - 1 + y + z = - 1 , x + α y - 1 + z = - 1 and x + y + α z - 1 = - 1 has no solution, if α is equal to

Question

The system of equations α x - 1 + y + z = - 1 , x + α y - 1 + z = - 1 and x + y + α z - 1 = - 1 has no solution, if α is equal to, - 2 o r 1, - 2, 1, - 1

MARKS App · Accepted Answer

Rearranging the given equations, we get, α x + y + z = α - 1 x + α y + z = α - 1 x + y + α z = α - 1 For no solution, Δ = 0 and atleast one of Δ 1 , Δ 2 , Δ 3 must be non zero. Δ = α 1 1 1 α 1 1 1 α = α α 2 - 1 - 1 α - 1 + 1 1 - α = α - 1 α 2 + α - 1 - 1 = 0 ⇒ α + 2 α - 1 2 = 0 ⇒ α = - 2 , 1 Δ 1 = α - 1 1 1 α - 1 α 1 α - 1 1 α = α - 1 α 2 - 1 - 1 α - 1 2 - 1 α - 1 2 = α - 1 2 α + 1 - 2 = α - 1 3 Δ 2 = α - 1 3 = Δ 3 for no solution α = - 2

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