Let A be a real-valued square symmetric matrix of rank 2 with sum_{i=1}^{3} sum_{j=1}^{3} A_{ij}^2 = 50. Consider the fo...

Question

Let A be a real-valued square symmetric matrix of rank 2 with sum_{i=1}^{3} sum_{j=1}^{3} 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?
A. Both I and II
B. I only
C. II only
D. Neither I nor II

Accepted Answer

B. I only A is a 3x3 real symmetric matrix of rank 2, so one eigenvalue is exactly 0. The sum of squares of all entries equals 50. For a symmetric matrix, the sum of squares of all entries equals the sum of squares of eigenvalues (trace of A^2 = sum of eigenvalues^2). So lambda_1^2 + lambda_2^2 + lambda_3^2 = 50 with lambda_3 = 0, giving lambda_1^2 + lambda_2^2 = 50. Statement I is correct: if both nonzero eigenvalues had |lambda| > 5, their sum of squares would exceed 50 + something, contradiction. Statement II is not necessarily true because we could have lambda_1 = lambda_2 = 5 exactly, meaning the largest magnitude equals 5, not strictly greater than 5.