Question Bank • QB_EM_ADV_024 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Solutions of linear equations

Question Statement:

A homogeneous system of linear equations $A X = 0$ with $m$ equations and $n$ variables where $m < n$ always has:

Options:

  • (A) Only the trivial solution $X = 0$
  • (B) Infinitely many non-trivial solutions
  • (C) No solution
  • (D) A unique non-zero solution

Answer Key:

B

Step-by-Step Solution & Explanation

For any homogeneous linear system $A X = 0$: 1. The trivial solution $X = 0$ always exists (the system is unconditionally consistent). 2. The rank of $A$ satisfies $r \le \min(m, n) = m < n$. 3. The nullity (dimension of solution space) is $n - r \ge n - m > 0$. Since there is at least one free variable, there are infinitely many non-trivial solutions.
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