Math miscellaneousHard
Question
The system of equations
αx + y + z = α - 1,
x + αy + z = α - 1,
x + y + αz = α - 1
has no solution, if α is
αx + y + z = α - 1,
x + αy + z = α - 1,
x + y + αz = α - 1
has no solution, if α is
Options
A.-2
B.either - 2 or 1
C.not -2
D.1
Solution
αx + y + z = α - 1
x + αy + z = α - 1
x + y + zα = α - 1
ᐃ =
= α(α2 - 1) - 1(α - 1) + 1(1 - α)
= α (α - 1) (α + 1) - 1(α - 1) - 1(α - 1)
⇒ (α - 1)[α2 + α - 1 - 1] = 0
⇒ (α - 1)[α2 + α - 2] = 0
[α2 + 2α - α - 2] = 0
(α - 1) [α(α + 2) - 1(α + 2)] = 0
(α - 1) = 0, α + 2 = 0 ⇒ α = - 2, 1; but α ≠ 1.
x + αy + z = α - 1
x + y + zα = α - 1
ᐃ =
= α(α2 - 1) - 1(α - 1) + 1(1 - α)
= α (α - 1) (α + 1) - 1(α - 1) - 1(α - 1)
⇒ (α - 1)[α2 + α - 1 - 1] = 0
⇒ (α - 1)[α2 + α - 2] = 0
[α2 + 2α - α - 2] = 0
(α - 1) [α(α + 2) - 1(α + 2)] = 0
(α - 1) = 0, α + 2 = 0 ⇒ α = - 2, 1; but α ≠ 1.
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