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Let M be a real 4 × 4 matrix. Consider the following statements:
S1: M has 4 linearly independent eigenvectors.
S2: M has 4 distinct eigenvalues.
S3: M is non-singular (invertible).
Which one among the following is TRUE?
- S1 implies S2
- S1 implies S3
- S2 implies S1
- S3 implies S2
Correct answer: S2 implies S1
Solution
If a matrix has 4 distinct eigenvalues, it guarantees that there are 4 linearly independent eigenvectors corresponding to those eigenvalues, as distinct eigenvalues lead to independent eigenvectors.
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