GATE | GATE-CS-2017 (Set 1) | Question 45

Let A be m×n real valued square symmetric matrix of rank 2 with expression given below.
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 engenvalues of A is/are necessarily CORRECT?
(A) Both (i) and (ii)
(B) (i) only
(C) (ii) only
(D) Neither (i) nor (ii)

Answer: (B)

Explanation: As a rank of A matrix = 2, hence

=> n-2 eigen values are zero. Let \lambda_1, \lambda_2, 0, 0 be the eigen values.
Given that \sum_{i=1}^{n} \sum_{j=1}^{n} A_{ij}^2 = 50————(1)

We know that \sum_{i=1}^{n} \sum_{j=1}^{n} A_{ij}^2 = Trace of (AA)T = Trace of A2 (since A is symetric) = \lambda_1^2 + \lambda_2^2+0+0 ————(2)

From (1) and (2) :
\lambda_1^2 + \lambda_2^2=50

Hence, atleast one of the given eigen values lies in [-5,5](only 1 is correct).

This solution is contributed by Sumouli Chaudhary.

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