GATE AG 2018 • Q36 MCQ • 2 Marks

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Matrices and determinants

Question Statement:

Rank of the matrix $\begin{bmatrix} 5 & 3 & -3 & -1 \\ 3 & 2 & -2 & -1 \\ 2 & -1 & 2 & 8 \end{bmatrix}$ is

Options:

  • (A) 1
  • (B) 2
  • (C) 3
  • (D) 4

Official Answer Key:

C

Step-by-Step Solution & Explanation

Order is 3 × 4, so rank ≤ 3. R₁ − R₂ = [2, 1, −1, 0]. Compare with R₃ = [2, −1, 2, 8] — not proportional, so R₃ is independent of the span of R₁, R₂. Check a 3 × 3 minor, columns 1, 2, 4: |5 3 −1; 3 2 −1; 2 −1 8| = 5(16 − 1) − 3(24 + 2) + (−1)(−3 − 4) = 75 − 78 + 7 = 4 ≠ 0. A non-zero 3rd-order minor exists ⇒ rank = 3.

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