GATE EC 2018 Set 1 — Question 21
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Let be a real matrix. Consider the following statements:S1: has 4 linearly independent eigenvectors.
S2: has 4 distinct eigenvalues.
S3: is non-singular (invertible).Which one among the following is TRUE?
S2: has 4 distinct eigenvalues.
S3: is non-singular (invertible).Which one among the following is TRUE?
Correct answer
(C) S2 implies S1
Solution
Let's evaluate the implications:
- S2 implies S1: A fundamental theorem in linear algebra states that if an matrix has distinct eigenvalues, it is guaranteed to have linearly independent eigenvectors (and is thus diagonalizable). For a matrix, 4 distinct eigenvalues 4 LI eigenvectors. So, S2 S1 is TRUE.
- S1 implies S2: A matrix can have 4 LI eigenvectors even if eigenvalues are repeated (e.g., the Identity matrix has 4 LI eigenvectors but only one distinct eigenvalue, 1). So, S1 S2 is FALSE.
- S1 implies S3: A matrix can be diagonalizable (have 4 LI eigenvectors) and still be singular (e.g., a zero matrix has 4 LI eigenvectors but is singular). So, S1 S3 is FALSE.
- S3 implies S2: An invertible matrix can have repeated eigenvalues (e.g., is invertible but has repeated eigenvalues). So, S3 S2 is FALSE.
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