GATE AG 2023 • Q11 MCQ • 1 Mark

Section 1: Engineering Mathematics

Linear Algebra

Subtopic: Matrices and determinants

Question Statement:

If A and B are square matrices of order 3 such that |A| = −1, |B| = 3, then |3AB| equals

Options:

  • (A) -81
  • (B) -27
  • (C) -9
  • (D) 81

Official Answer Key:

A

Step-by-Step Solution & Explanation

For an n×n matrix, |kM| = kⁿ|M|. Here n=3, so |3AB| = 3³×|AB| = 27×|A||B| = 27×(−1)×3 = −81. Answer: (A) -81. *Distractor note:* (B) -27 omits the |A||B|=−3 factor; (C) -9 wrongly uses 3¹ instead of 3³; (D) 81 drops the sign from |A|=−1.

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