GATE | Sudo GATE 2020 Mock I (27 December 2019) | Question 53

Which of the following matrices has similar eigenvalues ?



(A) Only A and B
(B) Only B and C
(C) Only A and C
(D) None of these


Answer: (B)

Explanation: The eigenvalues of the first matrix (A) are given by,

λ2 − (tr A)λ + det A = 0
⇒ λ2 − 4λ + 3 = 0
⇒ λ = 1, 3 

And the eigenvalues of the second matrix (B) are,

λ2 − (tr B)λ + det B = 0
⇒ λ2 − 4λ + 4 = 0 =
⇒ λ = 2, 2 

The other two matrices are lower triangular, so their eigenvalues are their diagonal entries.
This means that B, C are the only two matrices which has same eigenvalues, i.e., 2, 2.

Option (B) is correct.


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