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
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.