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

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  • Last Updated : 17 Aug, 2021
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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?


Both (i) and (ii)


(i) only


(ii) only


Neither (i) nor (ii)

Answer: (B)


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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