GATE CS 2017 Set 1 — Question 31
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Engineering Mathematics → Linear Algebra → Eigenvalues & Eigenvectors
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Question
Let be real valued square symmetric matrix of rank 2 with . Consider the following statements.I. One eigenvalue must be in
II. The eigenvalue with the largest magnitude must be strictly greater than 5Which of the above statements about eigenvalues of is/are necessarily CORRECT?
II. The eigenvalue with the largest magnitude must be strictly greater than 5Which of the above statements about eigenvalues of is/are necessarily CORRECT?
Correct answer
(B) (I) only
Solution
Given that is a symmetric matrix, the sum of the squares of its elements is equal to the sum of the squares of its eigenvalues (Frobenius norm property for symmetric matrices):Since the rank of is 2, there are exactly 2 non-zero eigenvalues, say and . The remaining eigenvalues are 0.
Thus, .Statement I: One eigenvalue must be in .
If , then 0 is an eigenvalue, and .
If , we have . Suppose both eigenvalues are outside . Then and , which implies and . Summing these gives , a contradiction. Therefore, at least one eigenvalue must satisfy , i.e., . Statement I is TRUE.Statement II: The eigenvalue with the largest magnitude must be strictly greater than 5.
Consider the case where and . Then . The largest magnitude is 5, which is not strictly greater than 5. Thus, Statement II is NOT necessarily true.Therefore, only Statement I is necessarily correct.
Thus, .Statement I: One eigenvalue must be in .
If , then 0 is an eigenvalue, and .
If , we have . Suppose both eigenvalues are outside . Then and , which implies and . Summing these gives , a contradiction. Therefore, at least one eigenvalue must satisfy , i.e., . Statement I is TRUE.Statement II: The eigenvalue with the largest magnitude must be strictly greater than 5.
Consider the case where and . Then . The largest magnitude is 5, which is not strictly greater than 5. Thus, Statement II is NOT necessarily true.Therefore, only Statement I is necessarily correct.
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