The following system of equations x₁ + x₂ + 2x₃ = 1 x₁ + 2x₂ + 3x₃ = 2 x₁ +…
2008
The following system of equations
x₁ + x₂ + 2x₃ = 1
x₁ + 2x₂ + 3x₃ = 2
x₁ + 4x₂ + αx₃ = 4
has a unique solution. The only possible value(s) for α is/are
- A.
either 0 or 1
- B.
any real number
- C.
0
- D.
any real number other than 5
Attempted by 2 students.
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Correct answer: D
A system of three linear equations in three variables has a unique solution when the determinant of its coefficient matrix is nonzero.
The coefficient matrix is
[1 1 2]
[1 2 3]
[1 4 α].
Its determinant is:
1(2α - 12) - 1(α - 3) + 2(4 - 2)
= 2α - 12 - α + 3 + 4
= α - 5.
For a unique solution, determinant ≠ 0.
So α - 5 ≠ 0, i.e. α ≠ 5.
Therefore, α can be any real number other than 5.