GATE 2017 CS – Question 41
Let $A$ be $n \times n$ real valued square symmetric matrix of rank 2 with $\sum_{i=1}^{n}\sum_{j=1}^{n} A_{ij}^2 = 50$. Consider the following statements.
(I) One eigenvalue must be in $[-5, 5]$
(II) The eigenvalue with the largest magnitude must be strictly greater than 5
Which of the above statements about eigenvalues of $A$ is/are necessarily CORRECT?
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Correct answer: (B) (I) only
Explanation
The sum of the squares of all entries equals the sum of the squares of the eigenvalues, so $\lambda_1^2 + \lambda_2^2 = 50$ for the two nonzero eigenvalues. The smaller one has $\lambda^2 \leq 25$, so some eigenvalue lies in $[-5, 5]$, and (I) is true. The larger has $\lambda^2 \geq 25$, but it can equal exactly 5 when both are $\pm 5$, so it need not be strictly greater than 5, and (II) is false.