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30. a, oe. a. Sey 36. Bik 38. ae, 40. 41. 42. 43. 44, 45. MATHEMATICAL PHYSICS:SDC 3 ee The eigenvalues of the inatix (2 Ae 7 16 9) are (A) 1 and —1; (B) 16 and 16; (C) 16 and —16; (D) 1 and 256; {Iisc,2007] Ge A 6 Consider the matrix ; : ; ; . The eigenvalues are Le Op (A) (1, —4, ~1, 4}; (B) {1, -1, 0,0}; © {1,1, 1,1, }; ) {1, 1,4, —é}; [l1sc,2008} i: P be an (n x n) diagonalizable matrix, which satisfies the equation P? = P and Tr[P] = n — 1; et P =? (A) n; (B) 0; (C) 1; (D) n — 1; [Iusc,2008] Let M be a (3 x 3) Hermitian matrix which satisfies the matrix equation M? — 5M + 61 = 0, where I refers to the unit matrix. Which of the following are possible eigenvalues of M? (A) {1, 2, 3}; (B) {2, 2, 3}; (C) {2, 3, 5}; (D) {5,5, 6}; [Iisc,2009] The trace of a (3 x 3) matrix is 1 and determinant is 0. Which of the following has to be true? (A) One of the eigenvalues is 0: (B) One of the eigenvalues is 1; (C) Two of the eigenvalues are 0; (D) Two of the eigenvalues are real; [IISc,201 0] 1 O -i 1 0 Given the three matrices 1, = G ie = € 5) 193 = c a and [Ou;, 5] —= 07104 — 079%, then [o1, [o2, a3] as [o2, [o3, o1]] alr [o3, lon, 72]| Is (A) of + 03 + 03; (B) 01 + 02 + 03; (C) 0; (D) identity; [1isc,2010] Let M be a (3 x 3) Hermitian matrix which satisfies the matrix equation M? — 7M + 121 = 0, where I refers to the identity matrix. What is the determinant of the matrix M given that the trace is 10? (A) 27; (B) 36; (C) 48; (D) 64; [Iisc,2011] The trace of a (2 x 2) matrix is 1 and its determinant is 1. Which of the following has to be true? (A) One of the eigenvalues is 0; (B) One of the eigenvalues is 1; (C) Both of the eigenvalues are 1: (D) Neither of the eigenvalues are 1; [I1sc,201 1] Given the three matrices g, = r 5) 102 = e 9)*23 = & si which of the following statements is true fcr all positive integers n and i = 1,2, 3 (A) oj} = I; (B) of = a;; (C) gee =a (i1B))) get = o;; [IISc,2011] a? aj,ag a,ag Consider a (3 x 3) matrix of the form | a,a2 as az =] . The number of zero eigenvalues for this a1a3 a2az3 az matrix is (Aa) 0; (3) 1; (C) 2: (D) 3: ise 2012) OP Oe 4 Given the three matrices T,, = C0 | ae, — teh). () | ee Te oN a} (5), 6} SS SS | ~. © Wits Re | EE ee | OS va ea which of the following statements is true? eet te + 0? = I (B) T? + ae Ol (CYT? +: ities (ey spee T?+T? =0; (lisc,2012] The trace and the determinant of a (3 x 3) matrix A satisfy Tr[A] = 2det A = 2. Further, the sum of two of the eigenvalues of A is equal to the third ei genvalue. Then the trace and the determinant of the matrix A* are, respectively (A) 2 and 1; (B) 0 and 1; (C) 4 and 1; (D) 4 and 2; (E) 4 and 4: [JNU,2015] The exponential of a (2x 2) matrix A is defined by the power series expansion exp(A) =) an n=0 nl with A® being the identity matrix. If S = mie i 5) and SAS~! = é ag then the determinant 3 of the matrix e4 is (A) 1; (B) e; (C) 2; (D) e + 4; (E) -1; [Jnu,2014] A homogeneous linear transformation takes the point (1,1) in the Z-y plane to the point (3,3) and keeps the point (1, —1) fixed (i.e., it remains (1,—1) after the transformation). The matrix corte- sponding to this transformation is i oN a 0N. By us A)(, 7): @(} 8): ee 2); UNu,2012) For which of the following matrices both the eigenvalues are positive? —7 aN, its oe ~ mt oe) eo ( 5) @) (~} 4); wu,2011) The determinant of (3 x 3) real symmetric matrix is 40 . etern If two of its eigenvalues are 2 and 4, then the third eigenvalue is 
