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GATE 2025 AE – Question 12

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

$2x + 3y + z = 0$

$x + y = 0$

$y + z = 0$

The given system of equations has

  1. a unique solution.
  2. infinitely many solutions.
  3. no solution.
  4. a finite number of solutions.

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Show answer and explanation

Correct answer: (B) infinitely many solutions.

Explanation

From the second and third equations $y = -x$ and $z = -y = x$. The first equation then gives $2x - 3x + x = 0$, which holds for every $x$. The three equations are not independent (the determinant is zero), and the solutions $(x, -x, x)$ are infinite in number.