Question

Let A be n x n real valued square symmetric matrix of rank 2 with \(\displaystyle\sum_{i=1}^{n}\) \(\displaystyle\sum_{j=1}^{n} A_{ij}^{2}\) = 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 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)

Answer 1

(B) (I) only 

p(A) < n ⇒ | A| = 0 ⇒ one eigen value must be '0'∈ [-5,5]

∴ (I) is true

but eigen values of A are 0, −5,5

∴ The eigen value with the largest magnitude is not greater than 5

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

∴ (II) is false