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GATE 2024 AE – Question 11

Engineering Mathematics · Linear Algebra: Vector algebra, matrix algebra, systems of linear equations, rank, eigenvalues and eigenvectors · 1 mark · Multiple choice

The following system of linear equations

$7x - 3y + z = 0$

$3x - y + z = 0$

$x - y - z = 0$

has:

  1. infinitely many solutions
  2. a unique solution
  3. no solution
  4. three solutions

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Show answer and explanation

Correct answer: (A) infinitely many solutions

Explanation

The system is homogeneous, so it always has the solution $x = y = z = 0$. The determinant is $7(1 + 1) + 3(-3 - 1) + 1(-3 + 1) = 14 - 12 - 2 = 0$, so there are non-trivial solutions as well, and so infinitely many.