Consider the following system of equations in three real variables x₁, x₂ and…
2005
Consider the following system of equations in three real variables x₁, x₂ and x₃:
2x₁ - x₂ + 3x₃ = 1
3x₁ - 2x₂ + 5x₃ = 2
-x₁ + 4x₂ + x₃ = 3
This system of equations has
- A.
no solution
- B.
a unique solution
- C.
more than one but a finite number of solutions
- D.
an infinite number of solutions
Attempted by 4 students.
Show answer & explanation
Correct answer: B
Write the coefficient matrix as
A = [[2, -1, 3], [3, -2, 5], [-1, 4, 1]].
The determinant is
|A| = 2[(-2)(1) - (5)(4)] - (-1)[(3)(1) - (5)(-1)] + 3[(3)(4) - (-2)(-1)]
= 2(-22) + 8 + 3(10)
= -44 + 8 + 30
= -6.
Since det(A) is non-zero, the coefficient matrix is invertible. Therefore the system has a unique solution.