Question

Let A be n x 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 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 statementss about eigen values of A is/are necessary CORRECT?

• A. Both (I) and (II)
• B. (I) Only
• C. (II) only
• D. Neither (I) nor (II)

Answer 1

The correct option is B (I) Only

For n > 2, $\rho$(A) = 2 and $A_{nxn}$

i.e., $\rho$(A) < n $\Rightarrow$ |A| = 0 one eigen value must be '0' $\epsilon$ [-5, 5]

$\therefore$ (1) is true

For n = 2 and let A = $\left[\begin{array}{cc}5& 0\\ 0& 5\end{array}\right]$

$\Rightarrow$ eigen values = 5, 5

One eigen value must be in [-5, 5] and largest eigen value magnitude is not greater than 5.

(II) is false