The GATE Grind

GATE 2018 EC – Question 21

Engineering Mathematics · Linear Algebra · 1 mark · Multiple choice

Let $\mathbf M$ be a real $4\times4$ matrix. Consider the following statements:

S1: $\mathbf M$ has 4 linearly independent eigenvectors.

S2: $\mathbf M$ has 4 distinct eigenvalues.

S3: $\mathbf M$ is non-singular (invertible).

Which one among the following is TRUE?

  1. S1 implies S2
  2. S1 implies S3
  3. S2 implies S1
  4. S3 implies S2

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Show answer and explanation

Correct answer: (C) S2 implies S1

Explanation

Eigenvectors belonging to distinct eigenvalues are linearly independent, so 4 distinct eigenvalues give 4 independent eigenvectors (S2 implies S1). The converse fails (the identity matrix), and a matrix with zero as an eigenvalue can still have 4 independent eigenvectors, so S1 does not imply S3.