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 eigenvalues 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 be the eigen values. Given that ————(1) We know that Trace of (AA)^T = Trace of A² (since A is symetric) = ————(2) From (1) and (2) : 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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