Question Bank • QB_SUB_EM_0020 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Solutions of linear equations

Question Statement:

A non-homogeneous system of linear equations $A x = b$ with $n$ variables is consistent and has a UNIQUE solution if and only if:

Options:

  • (A) $\text{rank}(A) = \text{rank}([A|b]) = n$
  • (B) $\text{rank}(A) < \text{rank}([A|b])$
  • (C) $\text{rank}(A) = \text{rank}([A|b]) < n$
  • (D) $\det(A) = 0$

Answer Key:

A

Step-by-Step Solution & Explanation

By Rouché-Capelli theorem, a linear system is consistent iff $\text{rank}(A) = \text{rank}([A|b])$. The solution is unique if this rank equals the number of variables $n$.
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