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GATE 2014 EE (EE1) – Question 11

Engineering Mathematics · Linear Algebra: Systems of linear equations · 1 mark · Multiple choice

Consider $x+2y+2z=b_1$ and $5x+y+3z=b_2$. Which statement about solutions is true?

  1. A unique solution exists for every b₁,b₂.
  2. Infinitely many solutions exist for every b₁,b₂.
  3. Existence of a solution depends on b₁,b₂.
  4. No solution exists for any b₁,b₂.

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Show answer and explanation

Correct answer: (B) Infinitely many solutions exist for every b₁,b₂.

Explanation

The coefficient rows(1,2,2)and(5,1,3)are linearly independent, so the2×3matrix has full row rank2. Its map onto the two-dimensional right-hand-side space is surjective:every(b1,b2)has a solution. One variable remains free, giving **infinitely many solutions for every right-hand side,B**.