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2 MATHEMATICAL PHYSICS -SDC eee ce is; 16. 17; 18. Ne) 20. pile De. DO): 24. 26. Dh 28. 29: . Given a matrix = ' ;) which of the following represents cos [| ? ing eigenvectors can If the eigenvalues of a symmetric (3 x 3) matrix A are 0, 1, 3 and the corresponding elg be written as J 1 t 1 0 —2 1 —1 f respectively, then the matrix A? is iv 27 481 ap 82 81°79 ee ON. D a 5 Sane ||: Sete 0 ei (By 81 a1 a1 |-(C) | 27. 54. 27): D) ee 40 -81 Al i Bi 83 13 —27 14 [TIFR,2016] The eigenvalues of the matrix representing the following pair of linear equations x + 7y = O and x+y =Oare (A) {1 + 2,1 + 7}; (B) {1 — 7,1 — i}; © {1,2}; (D) {1 44,1 — 4}. [JAM,2016] | | Two matrices A and B are said to be similar if B = P~1 AP for some invertible matrix P. Which of the following statements is NOT TRUE? (A) det A = det B; (B) Tr[A] = Tr[B]; (C) A and B have the same eigenvectors; (D) A and B have the same eigenvalues. [GATE,201 1] A (3 x 3) matrix has elements such that its trace is 11 and its determinant is 36. The eigenvalues of the matrix are all known to be positive integers. The largest eigenvalue of the matrix is (A) 18; (B) 12; (C) 9; (D) 6. [GATE,2011] fhe ete: The matrix A = Ja ( ey) is /3 \l—i —] (A) orthogonal; (B) symmetric; (C) anti-symmetric; (D) unitary. [GATE,2014] 1 oe The matrix M = (2 —1 7 Satisfies the equation 0 Os 1 (A) M* — M* — 10M +121 = 0; (B) M3 + M? — 12M + 101 = 0;(C) M3— M2—-10M +101 = 0: (D) M° + M* — 10M + 10I = 0; [NET,2016] x Evra Ty eles On, 08) 1.0,.1,.2}: (C) {1, 2, 3}; (D) {—1, 0, —2}; [Ism,2015] Define oz = (f' + f) ando, = —i(ft — f), where the o’s are Pauli spin matrices and f, f* satisfy {f, f} =0,{f, f"} =1. Then o, is given by (A) ff — 1; (B) 2f"f — 1; (© 2f'f +1; D) ft f; YeEst,2012] For an (N x NV) matrix consisting of all ones, (A) all eigenvalues = 1; (B) all eigenvalues = 0; (C) the eigenvalues are 2,5 ev (DD) one eigenvalue = NV, the others = 0; [JEST,2012] What are the eigenvalues of the operator H vector? (A) dz + ay and a,;(B) az + az + iay; (C) +(a, + dy + az); (D) +|al; [JEST,2013] De (lee One -0 Eigenvaiues of the matrix M@ = | OP] -are 0 = @.a, where o are the three Pauli matrices and a isa 6 (A) 2 é af (yee fi an Qe f a (D) 2 2 Mer [JEST,2016] Which of the following statements is true for a Square matrix A? (A) If A* = 0, it necessarily implies A = 0: (B) i always be real; (C) If A is Hermitian, its diagonal el its diagonal elements are always zero; [DU,201 3) The trace of a (4 x 4) matrix is 8 and its two eigenvalues are (A) —2 and 1; (B) 2 and —1; (C) 2 and (D>) —2 and cet UO Si) If the trace and the determinant of a (2 x 2) matrix are both zero then, (A) only one of eigenvalues needs to be zero eigenvalues are non-zero but finite: determinant is —24. If two eigenvalues are 3 and 4, the other ; (B) both the eigenvalues need to be Zero; (C) both the 3 21 1 The matrix | —2: 4 1 + 2% 1 1-2 9D (A) has one real and one complex eigenvalues: (B) i _ has three real eigenvalues; (D) has three complex eigenvalues: [Du,2014] 
