Let A be n × n real valued square symmetric matrix of rank 2 with \[\sum\limits_{{\text{i}} = 1}^{\text{n}} {\sum\limits_{{\text{j}} = 1}^{\text{n}} {{\text{A}}_{{\text{ij}}}^2} } = 50.\] Consider the following statements. I. One eigen value must be in [-5, 5] II. The eigen value with the largest magnitude must be strictly greater than 5. Which of the above statements about eigen values of A is/are necessarily CORRECT? A. Both I and II B. I only C. II only D. Neither I nor II

The correct answer is: B. I only

A real symmetric matrix has real eigenvalues. The sum of the eigenvalues of a real symmetric matrix is equal to the trace of the matrix, which is equal to the sum of the diagonal elements. In this case, the trace of $A$ is $50$. Since $A$ has rank 2, it has two nonzero eigenvalues. Therefore, the sum of the two nonzero eigenvalues of $A$ is $50$. This means that one of the eigenvalues must be in the range $[-50, 50]$. However, it is not necessarily the case that the eigenvalue with the largest magnitude must be strictly greater than 5. For example, if $A$ is the matrix $$A = \begin{bmatrix} 10 & 0 \\ 0 & -10 \end{bmatrix},$$ then the eigenvalues of $A$ are $10$ and $-10$, and the eigenvalue with the largest magnitude is $10$.

Here is a more detailed explanation of each option:

• Option A: Both I and II. This is not necessarily true. For example, if $A$ is the matrix $$A = \begin{bmatrix} 10 & 0 \\ 0 & -10 \end{bmatrix},$$ then the eigenvalues of $A$ are $10$ and $-10$, and the eigenvalue with the largest magnitude is $10$. However, the eigenvalue with the smallest magnitude is not in the range $[-5, 5]$.
• Option B: I only. This is true. As explained above, the sum of the eigenvalues of a real symmetric matrix is equal to the trace of the matrix, which is equal to the sum of the diagonal elements. In this case, the trace of $A$ is $50$. Since $A$ has rank 2, it has two nonzero eigenvalues. Therefore, the sum of the two nonzero eigenvalues of $A$ is $50$. This means that one of the eigenvalues must be in the range $[-50, 50]$.
• Option C: II only. This is not necessarily true. For example, if $A$ is the matrix $$A = \begin{bmatrix} 10 & 0 \\ 0 & -10 \end{bmatrix},$$ then the eigenvalues of $A$ are $10$ and $-10$, and the eigenvalue with the largest magnitude is $10$. However, the eigenvalue with the largest magnitude is not strictly greater than 5.
• Option D: Neither I nor II. This is not necessarily true. For example, if $A$ is the matrix $$A = \begin{bmatrix} 10 & 0 \\ 0 & -10 \end{bmatrix},$$ then the eigenvalues of $A$ are $10$ and $-10$, and neither eigenvalue is in the range $[-5, 5]$.