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Dips ienneny, , SAREE, MATHEMATICS & STATISTICS ——————, , , , Unit-Il: LINEAR ALGEBRA - 2.3 Matrices and Their Properties | 223 |, , CHAPTER, , 2.3, , Matrices and Their Properties, , , , , , , , Weel Multiple Choice Questions —_|, , 1. Let S={A:A[a,],., ay=0 or 1Vi,j,, , Xa, =1vi and Ya, =1vi ‘, 7 7, , Then the number of elements is Sis, , (a) 5, (b) 5°, (ec) 5!, (da) 55, , [June 2011 (3 Marks)], 2. Let Dbe a non-zero nxnreal matrix with, , n2>2. Which of the following implications is, valid?, , (a) det(D)=0 implies rank (D)=0, (b) det(D)=1 implies rank (D) 41, () rank (D)=1 implies det(D) #0, , @) rank (D)=n implies det(D) #1, , [June 2011 (3 Marks)], Let W be the vector space of all real, polynomials of degree at most 3. Define, e, , T:W —>W by (Tp) (x)= p'(x) where p’ is the, derivative of p. The matrix of Tin the basis, , {1, Kx, ey ,considered as column vectors,, , is given by, , 0000, 1, , (a) 0 00, 0020, 000 3, 0000, 1,00 0, , ®) 0200, 0030, 0100, 0020, , (c), 0003, 0000, 0123, 0000, , (@), 0000, 0000, , , , , , [June 2011 (3 Marks)], , , , © DIPS House, 28, Jia Sarai, Hauz Khas, Near I.I.T., New Delhi-110016 88-00-22-1386 #® 011-26537527, @ [email protected] www.dipsacademy.com, , Scanned with CamScanner

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CSIR-NET/JRF: Pure Mathematics ~ Previous Year Solved Paper [IS, , 4., , 6., , Ran Meoaasny, BN No.:978-81-937729-1-1] DD IPS ACADEMY, , , , , , The determinant of the matrix, , 100002, , 010020, , 001200),, , 002100|", , 020010, , 200001, , (a) 0, , ) -9, , () -27, , @ 1, , [Dec. 2011 (3 Marks)], , 40 -29 -11, , Suppose the matrix A=|-18 30 -12| has, 26 24 -50, , a certain complex number 40 as an, , eigenvalue. Which of the following numbers, must also be an eigenvalue of A?, , (a) 2+20, () 4-20, (c) 20-24, @) -20-a, , [Dec. 2011 (3 Marks)], , Let A be a 54 matrix with real entries such, that the space of all solutions of the linear, , system AX‘ =[1,2,3,4,5] is given by, {1+2s,2 +3s,34+48,4+ 5s]! :se R} 3, , (Here y* denotes the transpose of a matrix, M). Then the rank of A is equal to, , (a) 4, ) 3, (c) 2, @ 1, , [Dec. 2011 (3 Marks)], Let A be a 3x3 matrix with real entries such, , that det(A)=6 and the trace of A is 0. If, , det(A+J)=0, where I denotes the 3x3, identity matrix, then the eigenvalues of Aare, , 10., , (a) -1,2,3, (bo) -1,2,-3, () 1,2,-3, @ -1,-2,3, , [Dec. 2011 (3 Marks)}, Let A, Bbe nxn real matrices., Which of the following statement is correct?, , (a) Rank (A+B)= rank (A)+ rank (B), (b) Rank (A+B) < rank (A)+ rank (B), (c) Rank(A+B)=min {rank(A), rank (B)}, , (@) Rank(A+B)=max {rank(A), rank (B)}, : [June 2012 (3 Marks}], Let f(x) be the minimal polynomial of the, , 4x4 matrix A=, , coro, oroo, roo°o, coor, , Then the rank of the 4x4 matrix f(A)is, , (a) 0, ) 1, () 2, @ 4, , [Dec. 2012 (3 Marks)], Leta, b,c be positive real numbers such that, , b? +c? <a<1. Consider the 3x3 matrix, , 1b, A=|ba, ec 0, , rOo°, , (a), , All the eigenvalues of A are negative, real numbers, (b) All the eigenvalues of A are positive real, numbers, (c) Acan have a positive as well as a, negative eigenvalue, (a) Eigenvalues of A can be non-real, complex numbers, [Dec. 2012 (3 Marks)], , , , , , HOUSE, 28, Jia Sarai, Hauz Khas, Near I.1.T., New Delhi-110016 (@ 88-00-22-1386 ‘MP 011-26537527, @ DIPS @ [email protected] gy dew dipsacadenyseon, , Scanned with CamScanner

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Dips icanen, , ECE, MATHEMATICS & STATISTICS., , 11. The system of equations, x+y+z=1, 2x+3y-z=5, x+2y-—kz=4, , where keR, has an infinite number of, , solution for ,, , (a) k=0, , 0) k=1, , () k=2, , @ k=3, , [Dec. 2012 (3 Marks], 13 5a 13, , 12. Let A=|O 1 7 9 b |wherea,beR., 001 1115, , Choose the correct statement., , (a) There exist value of a and b for which, the columns of A are linearly, independent, , (b) There exist values of a and b for which, Ax=0 has x =Oas the only solution, , (c) For all values of a and b the rows of A, span a 3-dimensional subspace of R®, , (a) There exist values of a and b for which, rank (A) =2, , [June 2013 (3 Marks)], , 13. Let A be a 5x4 matrix with real entries such, , that Ax =0 if and only if x=0 where x isa, 4x1 vector and 0 is a null vector. Then, the, tank of A is, , (a) 4, , ® 5, , () 2, , @ 1, , [Dec. 2013 (3 Marks)], , 14., , Unit-Il: LINEAR ALGEBRA - 2.3 Matrices and Their Properties Eg, , , , , , Let Aun =((a;)), n>3, where a, =(b? -b}),, i,j =1,2,...,n for some distinct real numbers, b,,b,,...,b,. Then det(A) is, , @ TL,(%-4), , 15., , 16., , 17., , wo TMe(& +), () 0, @ 1, , [Dec. 2013 (3 Marks)], Let Abe a 4x4 invertible real matrix. Which, of the following is NOT necessarily true?, The rows of A form a basis of R*, Null space of A contains only the 0 vector, , Ahas 4 distinct eigenvalues, Image of the linear transformation, , (a), (b), (c), (@), , x Ax on R* is R*, [Dec. 2013 (3 Marks)], , Let M,,.,(R) be the set of all mxn matrices, , with real entries. Which of the following, statements is correct?, , (a) There exists A¢ M,,, (R) such that the, dimension of the null space of A is 2, , (b) There exists Ae M,,,(R) such that the, dimension of the null space of Ais 0., , (c) There exists AeM,,(R) and, BeM,,,(R) such that AB is the 2x2, identity matrix, , (@) There exists AeM,,,(R) whose null, , spaceis {(X,)Xp1%1X4o%s) €R°: x, =X,, , Xy = X4 = Xs}, [June 2014 (3 Marks)], For the matrix Aas given below, which of them, satisfy A° =I?, , cos— sin— 0, 4, (a) =|-sin= cos 0, 0 oO 1, 1 0 0, ) A=|0 cos2 sin=, 3, 0 -sin= =, ‘sinZ cosz, , , , © DIPS HOUSE, 28, Jia Sarai, Hauz Khas, Near I.1.T., New Delhi-110016 88-00-22-1386 ‘@ 011-26537527, ® [email protected] @ www.dipsacademy.com, , Scanned with CamScanner

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Dips inniny, Eq CSIR-NET/JRF: Pure Mathematics ~ Previous Year Solved Paper [ISBN No.:978-81-937729-1-1] Pp: ADEMY, , ———~ MATHEMATICS & STATIS Tey, , , , (a) Ais non-singular, _ * © sind (b) Bis non-singular, 6 6 (c) A+Bis non-singular, () A=| 0 12 « (@) ABis non-singular, . 2., -sing O cos 6 [Dec. 2014 (3 Marks)}, 22. Which ofthe following matrices has the same, , cost sinZ 0 484, ; S ‘= row space as the matrix|3 6 1]?, @ A= -sin> cosy 0 240, 0 0 1, [June 2014 (3 Marks)] (a) c : "\, 18. Let Jdenote a 101 x 101 matrix with all the, entries equal to 1 and let [denote the identity 110, matrix of order 101. Then the determinant of (b) c 0 ‘), J-Iis, (a) 101 () a 1 ‘, 6 1 0 1, () 0, (@) 100 @ ( 10 a, [June 2014 (3 Marks)] Od 2, 19. Let Abea5~ 5 matrix with real entries such [Dec. 2014 (3 Marks)], , that the sum of the entries in each row ofA 23, The determinant of the nxn permutation, is 1. Then the sum of all the entries in A® is, , (a) 3 \ 1, ® 15, ©, 8 matrix ., @ 125 : 3, [June 2014 (3 Marks)] 4, a6. Gi : “(12345 i, . Given the permutation o = 31254, owe (a) (-1", the matrix A is defined to be‘'the one whose, i” column is the o(i)” column of the identity (b) call, matrix I, Which of the following is correct? @ ‘, ce) =, =A?, asa @ 1, 0) A=aA*, 3 : Here |x| denotes the greatest integer not, a ana exceeding x,, @ Ana" [Dec. 2014 (3 Marks}], [June 2014 (3 Marks)], 21. Let A, B be nxnmatrices such that lL l+x l+x+x, BA+B? =I-BA? where lis the nxnidentity 24. Thedeterminant |l l+y 1+y+y"Jis equal to, matrix. Which of the followings always true? 1 1+z l+z+2?, , , , , , © DIPS HOUSE, 28, Jia Sarai, Hauz Khas, Near |.1.7., New Delhi-110016 88-00-22-1386 # 011-26537527, @ [email protected] By www.dipsacademy.com, , Scanned with CamScanner

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27., , Dips itiieny, , eee, MATHEMATICS & STATISTICS., , Unit-ll: LINEAR ALGEBRA - 2.3 Matrices and Thel, , ir Properties, , Se, , , , (a) (2-y)(z-x)\(y-x), () (x-y)(x-z)(y-z), () (x-yP(y-2P(e-x?, (@) (x? -y*)(y? - 27)(2? - x?), [Dec. 2014 (3 Marks)], Let Pbea 2x 2 complex matrix such that p * p, , is the identity matrix, where p* is the, , conjugate transpose of P. Then the, eigenvalues of P are, , (a) Real, , (b) Complex conjugates of each other, (c) Reciprocals of each other, , (da) ofmodulus 1, , 25., , [Dec. 2014 (3 Marks)], , The row space of a 20x50 matrix A has, dimension 13. What is the dimension of the, space of solutions of Ax =0 ?, (a) 7, (b) 13, (c) 33, @ 37, , [June 2015 (3 Marks)], Let A, B be nxn matrices. Which of the, following equals trace (AB?) ?, (a) (trace (AB), (b) .. trace(AB?A), (c) trace ((4B)’), (@) _ trace(BABA), , [June 2015 (3 Marks)], Given a 4x4 real matrix A, let T: R* > R*be, , the linear transformation defined by Tv = Av,, , where we think of R‘as the set of real 4*1, matrices. For which choices of A given below,, , do Image (T) and Image (7?) have respective, , dimensions 2 and 1? (*denotes a non-zero, entry), , 29., , 30., , 31., , , , , , foo * *], , oo. % £, , () A=|) 9 0 *, lo 0 o 0], , fo o * 0], 0o0* 0, , ® 4*l000*, lo 0 o *], , fo 0 0 Oj, aul? 222, (c) =loo00*, lo o * oO}, , fo 0 0 O], 0000, , @ A=|, 9 «+, [0 ‘o-*: *], , [June 2015 (3 Marks)], , Let A be an mxnmatrix of rank n with real, entries. Choose the correct statement, , (a) Ax=b has a solution for any b, (b) Ax =0 does not have a solution, (c) If Ax = bhasa solution, then it is unique, , (a) = y'A=0 for some non-zero y, where y’, denotes the transpose of the vector y, , . [June 2015 (3 Marks)], , Let A be a real 3x4 matrix of rank 2. Then, , the rank of A'A, where 4‘ denotes the, transpose A, is:, , (a) Exactly 2, (b) Exactly3, (c) Exactly 4, (a4) At most 2 but not necessarily 2, , [Dec. 2015 (3 Marks)], If Ais a 5x5 real matrix with trace 15 and if, 2 and 3 are eigenvalues of A, each with, algebraic multiplicity 2, then the determinant, of Ais equal to, , (a) 0, (o) 24, , () 120, (@ 180, , [Dec. 2015 (3 Marks)], , © DIPS HOUSE, 28, Jia Sarai, Hauz Khas, Near |.1.T., New Delhi-110016 (@ 88-00-22-1386 011-26537527, ‘ ® [email protected] [J www.dipsacademy.com, , Scanned with CamScanner