EXERCISE - 01 CHECK YOUR GRASP

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1 XRCIS - 1 CHCK YOUR GRASP SLCT TH CORRCT ALTRNATIV (ONLY ON CORRCT ANSWR) 1. If A B = and A 3B = 5 7, then matrix B is equal to (B) If A = cos sin sin cos, then A A is equal to - A + (B) A A none of these 3. If number of elements in a matrix is 6 then how man different order of matrix are possible - 1 (B) 6 4 none of these 4. Matrix A has x rows and x + 5 columns. Matrix B has rows and 11 columns. Both AB and BA exist, then - x = 3, = 4 (B) x = 4, = 3 x = 3, = 8 x = 8, = 3 5. If A = A, then(i + A) 4 is equal to - I + A (B) I + 4A I + 15A none of these n If the product of n matrices... is equal to the matrix of n is equal to - 6 (B) then the value 7. If A = 1 and (ai +ba) = A, then - a = b = (B) a = b = 1/ a = b = 3 a = b = 1/ 3 8. If A is a skew smmetric matrix such that A T A = I, then A 4n 1 n N 5 is equal to - A T (B) I I A T 9. If AA T = I and det = 1, then - ver element of A is equal to it's co-factor. (B) ver element of A and it's co-factor are additive inverse of each other. ver element of A and it's co-factor are multiplicative inverse of each other. None of these 1. Which of the following is an orthogonal matrix - 6 / 7 / 7 3 / 7 6 / 7 / 7 3 / 7 / 7 3 / 7 6 / 7 (B) / 7 3 / 7 6 / 7 3 / 7 6 / 7 / 7 3 / 7 6 / 7 / 7 6 / 7 / 7 3 / 7 6 / 7 / 7 3 / 7 / 7 3 / 7 6 / 7 / 7 / 7 3 / 7 3 / 7 6 / 7 / 7 6 / 7 / 7 3 / If A is an orthogonal matrix & A = 1, then A T is equal to - A (B) A (adj A) (adj A)

2 Let A = 1 3 and 1B = 5. If B is the inverse of matrix A, then is (B) Let the matrix A and B be defined as A = and B then the value of Det.(A9 B 1 ), is - (B) If A , then matrix A equals (B) If A = 1 6. If M = 5 and (x) = 1 + x + x x 16, then = 1 5 (B) 1 3 and M M I = O, then equals (B) If A = 3, B = and ABC = , then C equals (B) If P is a two-rowed matrix satisfing P T = P 1, then P can be - cos sin sin cos (B) cos sin sin cos a 1 9. If A a, then A Adj A is equal to - a. If cos sin sin cos none of these a 5 (B) a 7 a 81 none of these 3 A 5, then 19A 1 is equal to - A T (B) A 1 A A SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS) 1. If A and B are square matrices of same order, then which of the following is correct - A + B = B + A (B) A + B = A B A B = B A AB = BA. A square matrix can alwas be expressed as a sum of a smmetric matrix and skew smmetric matrix of the same order (B) difference of a smmetric matrix and skew smmetric matrix of the same order skew smmetric matrix smmetric matrix 51

3 3. Choose the correct answer : ever scalar matrix is an identit matrix. (B) ever identit matrix is a scalar matrix transpose of transpose of a matrix gives the matrix itself. for ever square matrix A there exists another matrix B such that AB = I = BA. 4. Let a ij denote the element of the i th row and j th column in a 3 3 matrix and let a ij = a ji for ever i and j then this matrix is an - orthogonal matrix (B) singular matrix matrix whose principal diagonal elements are all zero skew smmetric matrix 5. Let A be an invertible matrix then which of the following is/are true : 6. If A 1 = A 1 (B) (A ) 1 = (A 1 ) (A T ) 1 = (A 1 ) T none of these n 1 6 n A i 8, where i 1 and is complex cube root of unit, then tr will be- 1, if n = 3k, k N (B) 3, if n = 3k, k N, if n 3k, k N 1, if n 3k, n N 7. If A is a square matrix, then - AA T is smmetric (B) AA T is skew-smmetric A T A is smmetric A T A is skew smmetric. 8. If A = a b c d satisfies the equation x + k =, then - a + d = (B) k = A k = a + b + c + d k = A 9. If A and B are invertible matrices, which one of the following statement is/are correct - Adj = A A 1 (B) det(a 1 ) = det 1 3. Matrix (A + B) 1 = B 1 + A 1 (AB) 1 = B 1 A 1 a b a b b c b c 1 is non invertible if - = 1/ (B) a, b, c are in A.P. a, b, c are in G.P. a, b, c are in H.P. CHCK YOUR GRASP ANSWR KY X R CI S - 1 Que A ns. A A A C C B B D A A Que A ns. C D D A B D B B D D Que A ns. A A, B B, C B,C,D A,B,C B, C A,C A, D A,B,D A,C 5

4 XRCIS - BRAIN TASRS SLCT TH CORRCT ALTRNATIVS (ON OR MOR THAN ON CORRCT ANSWRS) 1. If A and B are square matrices of same order and AA T = I then (A T BA) 1 is equal to - AB 1 A T (B) A T B 1 A A 1 B 1 (A T ) 1 1A T BA. If A is a invertible idempotent matrix of order n, then adj A is equal to - (adj A) (B) A 1 none of these x 3 3. Matrix A = 1 4, if xz = 6 and 8x z =, then A (adj A) is equal to - z (B) Let three matrices A = ; B = and C = then 3 ABC A(BC) A(BC) t r (A ) tr tr t r (B) 9 1 none of these 5. Let A, B, C, D be (not necessaril square) real matrices such that A T = BCD ; B T = CDA; C T = DAB and D T = ABC for the matrix S = ABCD, then which of the following is/are true S 3 = S (B) S = S 4 S = S none of these 1 tan x 6. If A = tan x 1 then let us define a function f(x) = det (AT A 1 ) then which of the following can be the value of f(f(f(f...f(x)))) (n ) n times f n (x) (B) 1 f n 1 (x) nf(x) 5 1 r 1 7. For a matrix A =, the value of 1 r 1 r 1 is equal to (B) If A and B are two invertible matrices of the same order, then adj (AB) is equal to - adj (B) adj (B) B A B 1 A 1 B A A 1 B 1 A B (AB) 1 1 / 1 / 1 / 1 9. If A = 1 3, A 4 3 c, then - 3 a 1 5 / 3 / 1 / a = 1, c = 1 (B) a =, c = 1 a = 1, c = 1 a = 1, c = 1 1. If A and B are different matrices satisfing A 3 = B 3 and A B = B A, then which of the following is/are incorrect- det (A + B ) must be zero (B) det (A B) must be zero det (A + B ) as well as det (A B) must be zero At least one of det (A + B ) or det (A B) must be zero 53

5 1 1. If A, then- 1 AdjA is zero matrix A 1 = A (B) Adj A 1 A = I 1. If A and B are square matrices of the same order such that A = A, B = B, AB = BA, then which one of the following ma be true- A(B) = O (B) (A + B) = A + B (A B) = A B none of these 1 3. If B is an idempotent matrix and A = I B, then- A = A (B) A = I AB = O BA = O a a a Let = a1 a a3 (where ) and let 1 denote the determinant formed b the cofactors of a31 a3 a33 elements of and denote the determinant formed b the cofactor at 1 and so on n denotes the determinant formed b the cofactors at n 1 then the determinant value of n is - n (B) n n BRAIN TASRS ANSWR KY X R CI S - Que A ns. B A,B,C C A A, B A,B,C D A,B,D A A,B,C Que A ns. B,C,D A,B,C A,C,D B 54

6 XRCIS - 3 MISCLLANOUS TYP QUSTIONS TRU / FALS 1. Let A, B be two matrices such that the commute, then A B = (A B)(A + B). If A is a periodic matrix with period then A 6 = A. 3. Let A, B be two matrices such that the commute, then (AB) n = A n B n. 4. All positive odd integral powers of skew smmetric matrix are smmetric. 5. Let A, B be two matrices, such that AB = A and BA = B, then A = A and B = B. 6. If A & B are smmetric matrices of same order then AB BA is smmetric. 7. If A and B are square matrices of order n, then A and B will commute, iff A I and B I commute for ever scalar. MATCH TH COLUMN Following questions contains statements given in two columns, which have to be matched. The statements in Column-I are labelled as A, B, C and D while the statements in Column-II are labelled as p, q, r and s. An given statement in Column-I can have correct matching with ON statement in Column-II. 1. Column-I Column-II Mat r i x Tpe of matrix (p) Idempotent (B) (q) Involutar (r) Nilpotent (s) Orthogonal. Column-I Column-II If A is a square matrix of order 3 and (p) 6 A det A = 16 then det = 3 (B) If A is a matrix such that A = A and (q) 5 (I + A) 5 = I + A then 1 7 If A = and A xa + I = (r) then x = If A = B = and (s) 9 then (AB) 3 55

7 ASSRTION & RASON These questions contains, Statement I (assertion) and Statement II (reason). Statement-I is true, Statement-II is true ; Statement-II is correct explanation for Statement-I. (B) Statement-I is true, Statement-II is true ; Statement-II is NOT a correct explanation for statement-i Statement-I is true, Statement-II is false Statement-I is false, Statement-II is true 1. Statement - I : If a, b, c are distinct real number and x,, z are not all zero given that ax + b + cz =, bx + c + az =, cx + a + bz =, then a + b + c Statement - II : If a, b, c are distinct positive real number then a + b + c ab + bc + ca. A (B) B C D. Statement - I : If A is skew smmetric matrix of order 3 then its determinant should be zero Statement - II : If A is square matrix, then det A = det A' = det( A') A (B) B C D 3. Statement-I : If A is a non-singular smmetric matrix, then its inverse is also smmetric. B e c a u s e Statement-II : (A 1 ) T = (A T ) 1, where A is a non-singular smmetric matrix. A (B) B C D 4. Statement - I : There are onl finitel man matrices which commute with the matrix B e c a u s e Statement - II : If A is non-singular, then it commutes with I, adj A and A 1. A (B) B C D Statement-I : If x = A & if A is idempotent matrix then x is also idempotent matrix. B e c a u s e Statement-II : If P is an orthogonal matrix & Q = PAP T, then Q n = PA n P T. A (B) B C D 6. Statement-I : The determinants of a matrix A = [a ij ] 5 5 where a ij + a ji = for each i and j is zero. B e c a u s e Statement-II : The determinant of a skew smmetric matrix of odd order is zero. A (B) B C D 56

8 COMPRHNSION BASD QUSTIONS Comprehension # 1 Let P(x, ) be an point and P'(x 1, 1 ) be its image in x-axis then x 1 = x 1 = This sstem of equation is equivalent to the matrix equation. x1 x A 1 where A is a square matrix of order Similarl x x B, x3 x C, 3 x 4 x D 4 represents the reflection of point (x, ) in -axis, origin and the line = x respectivel. On the basis of above information, answer the following questions : 1. The value of A + B + C + D is (B) Let X be a square matrix given b X = A + AD + AD AD n, then X is - n n (B) n n n n n n 3. x a Let P(a, b) be a point & DCBA b then Q(x, ) represents the reflection of point P(a, b) in - x-axis (B) -axis origin line = x Comprehension # a1 a a3 Matrix A is called orthogonal matrix if AA T = I = A T A. Let A = b1 b b 3 be an orthogonal matrix. Let c1 c c 3 ˆ ˆ ˆ a a1i a j a3k, ˆ ˆ ˆ b b1i b j b3k, ˆ ˆ ˆ c c1i c j c3k. Then a b c 1 & a.b b.c c.a i.e. a, b & c forms mutuall perpendicular triad of unit vectors. a b c If abc = p and Q = c a b, where Q is an orthogonal matrix. Then. b c a On the basis of above information, answer the following questions : 1. The values of a + b + c is - (B) p p ±1. The values of ab + bc + ca is - (B) p p 3p 3. The value of a 3 + b 3 + c 3 is - p (B) p 3p None of these 4. The equation whose roots are a, b, c is - x 3 x + p = (B) x 3 px + px + p = x 3 x + px + p = x 3 ± x p = 57

9 Comprehension # If A and B B n = adj(b n 1 ), n N and I is an identit matrix of order 3. On the basis of above information, answer the following questions : det. (A A B A A B...1 terms) is equal to - 1 (B) 8 8. B 1 + B B 49 is equal to - B (B) 7B 49B 49I 3. For a variable matrix X the equation A X = B will have - unique solution (B) infinite solution finitel man solution no solution MISCLLANOUS TYP QUSTION ANSWR KY X R CI S - 3 Tr ue / False 1. T. F 3. T 4. F 5. T 6. F 7. T Match the Column 1. (p), (B) (q), (s), (r). (p), (B) (s), (q), (r) Assertion & Reason 1. D. C 3. A 4. D 5. A 6. A Comprehension Based Que st ions Comprehensi on # 1 : 1. B. C 3. D Comprehensi on # : 1. D. A 3. D 4. D Comprehensi on # 3 : 1. C. C 3. D 58

10 XRCIS - 4 [A] CONCPTUAL SUBJCTIV XRCIS 1. Find the value of x and that satisf the equations x x 4 1. A is a square matrix of order n. = maximum number of distinct entries if A is a triangular matrix m = maximum number of distinct entries if A is a diagonal matrix p = minimum number of zeroes if A is a triangular matrix If + 5 = p + m, find the order of the matrix. 3. If the matrices A = and B = a b c d (a, b, c, d not all simultaneousl zero) commute, find the value of / 3 commutes with A is of the form d b. Also show that the matrix which a c b 1 4. Consider the two matrices A and B where A = 4 3 ; B = 53. Let n denotes the number of elements in A. When the two matrices X and Y are not conformable for multiplication then n(xy) = If C = (AB)(B'A); D = (B'A)(AB) then, find the value of n ( D n n(a ) n(b). 5. Define A = 3. Find a vertical vector V such that (A8 + A 6 + A 4 + A + I)V = 11 (where I is the identit matrix). 6. If A is an idempotent matrix and I is an identit matrix of the same order, find the value of n, n N, such that (A + I) n = I + 17 A. 7. If the matrix A is involutar, show that 1 (I + A) and 1 (I A) are idempotent and 1 (I + A) Let X be the solution set of the equation A x = I, where A = and x N then find the minimum value of 9. xpress the matrix x x (cos sin ), R. (I A) =. and I is the corresponding unit matrix as a sum of a lower triangular matrix & an upper triangular matrix with zero in leading diagonal of upper triangular matrix. Also express the matrix as a sum of a smmetric and a skew smmetric matrix. 59

11 1. Given A = , B = Find P such that BPA = If A = sin cos cos sin then find A T and A 1. 1 tan 1. Show that, tan 1 1 tan tan 1 1 cos sin = sin cos cos x sin x If F(x) = sin x cos x then show that F(x).F() = F(x + ). Hence prove that [F(x)] 1 = F( x) If A = 1, then show that the matrix A is a root of the polnomial f(x) = x 3 6x + 7x Use matrix to solve the following sstem of equations (a) x z 3 x 3z 4 x 4 9z 6 (b) x z 6 x z x z 1 (c) x z 3 x 3z 4 x 3 4z 7 (d) x z 3 x 3z 4 x 3 4z 9 CONCPTUAL SUBJCTIV XRCIS ANSWR KY X R C IS - 4 ( A ) 1. x = 3/, = ; V n = (a) x =, = 1, z = (b) x = 1, =, z = 3 (c) x = + k, = 1 k, z = k where k R (d) inconsistent, hence no solution 11. 1, 1

12 XRCIS - 4 [B] BRAIN STORMING SUBJCTIV XRCIS 1. If A = a b c d then prove that value of f and g satisfing the matrix equation A + fa + gi = O are equal to 1 t r and determinant of A respectivel. Given a, b, c, d are non zero reals and I = ; O =.. A 3 3 is a matrix such that A =a, B = (adj A) such that B = b. Find the value of (ab + a b + 1)S where 3 1 S = a a a up to, and a = 3. b b b 3. Find the number of matrix satisfing : (a) a ij is 1 or 1 ; (b) a a a a ; (c) a 11 a 1 + a 1 a = If A is a skew smmetric matrix and I + A is non singular, then prove that the matrix B = (I A)(I + A) 1 is an 5 orthogonal matrix. Use this to find a matrix B given A = Given A = 1 ; B = I is a unit matrix of order. Find all possible matrix X in the following cases. (a) AX = A (b) XA = I(c) XB = O but BX O. 3 1 x b 6. Determine the values of a and b for which the sstem a z 1 (a) has a unique solution ; (b) has no solution and (c) has infinitel man solutions 7. If A is an orthogonal matrix and B = AP where P is a non singular matrix then show that the matrix PB 1 is also orthogonal. n 1 a If 4 36 then find a + n. 9. Let a A c b d and p P q. Such that AP = P and a + d = 55. Find the value of (ad bc). BRAIN STORMING SUBJCTIV XRCIS ANSWR KY X R C I S - 4 ( B ) a b 5. (a) X = a 1 b for a, b R ; (b) X does not exist ; (c) X = a 3a c 3c a, c R and 3a + c 6. (a) a 3 ; b R ; (b) a = 3 and b 1/3 ; (c) a = 3, b = 1/

13 XRCIS - 5 [A] J-[MAIN] : PRVIOUS YAR QUSTIONS a b 1. If A = b a and A = then [AI 3] (1) = a + b, = a b () = a + b, = ab (3) = a + b, = ab (4) = ab, = a + b. If A = 1 then- [AI 4] (1) A 1 does not exist () A = (3) A = (4) A = ( 1) 3. If A = and 1B = where B = A 1, then is equal to- [AI 4] (1) () 1 (3) (4) 5 4. If A A + I =, then the inverse of A [AI 5] (1) I A () A I (3) A (4) A + I L NM 5. If A = O L and I = QP 1 O NM QP, then which one of the following holds for all n1, (b the principal of mathematical induction) [AI-5] (1) A n = na (n 1) I () A n = n-1 A+ (n 1) I (3) A n = na + (n 1) I (4) A n = n-1 A (n 1) I 6. If A and B are square matrices of size n n such that A B = (A B) (A + B), then which of the following will be alwas true- [AI- 6] (1) AB = BA () ither of A or B is a zero matrix (3) ither of A or B is an identit matrix (4) A = B 7. Let A = and B = a, a, b N. Then- [AI- 6] b (1) there exist more than one but finite number of B's such that AB = BA () there exist exactl one B such that AB = BA (3) there exist infinitel man B's such that AB=BA (4) there cannot exist an B such that AB = BA Let A = 5 If A = 5, then equals- [AI- 6] 5 (1) 5 () 1 (3) 1/5 (4) 5 9. Let A be a matrix with real entries. Let I be the identit matrix. Denoted b tr, the sum of diagonal entries of A. Assume that A = I. Statement 1: If A I and A I, then det A = 1 [AI- 8] Statement : If A I and A I, then tr. (1) Statement 1 is false, Statement is true. () Statement 1 is true, Statement is true; Statement is a correct explanation for Statement 1. (3) Statement 1 is true, Statement is true; Statement is not a correct explanation for Statement 1. (4) Statement 1 is true, Statement is false 6

14 1. Let A be a square matrix all of whose entries are integers. Then which one of the following is true? [AI- 8] (1) If det A = ± 1, then A 1 exists but all its entries are not necessaril integers () If det A ±1, then A 1 exists and all its entries are non integers (3) If det A = ±1, then A 1 exists and all its entries are integers (4) If det A = ±1, then A 1 need not exist 1 1. Let A be a matrix [AI- 9] Statement 1 : adj (adj A) = A Statement : adj A = A (1) Statement 1 is true, Statement is false. () Statement 1 is false, Statement is true. (3) Statement 1 is true, Statement is true; Statement is a correct explanation for Statement 1. (4) Statement 1 is true, Statement is true; Statement is not a correct explanation for statement The number of 3 3 non-singular matrices, with four entries as 1 and all other entries as, is :- [AI-1] (1) Less than 4 () 5 (3) 6 (4) At least Let A be a matrix with non-zero entries and let A = I, where I is identit matrix. Define Tr = sum of diagonal elements of A and A = determinant of matrix A. [AI-1] Statement 1 : Tr =. Statement : A = 1. (1) Statement 1 is true, Statement is true; Statement is a correct explanation for Statement 1. () Statement 1 is true, Statement is true; Statement is not a correct explanation for statement 1. (3) Statement 1 is true, Statement is false. (4) Statement 1 is false, Statement is true Let A and B be two smmetric matrices of order 3. Statement-1 : A(BA) and (AB)A are smmetric matrices. Statement- : AB is smmetric matrix if matrix multiplication of A with B is commutative. (1) Statement-1 is true, Statement- is false. () Statement-1 is false, Statement- is true (3) Statement-1 is true, Statement- is true; Statement- is a correct explanation for Statement-1 [AI-11] (4) Statement-1 is true, Statement- is true; Statement- is not a correct explanation for Statement Statement-1 : Determinant of a skew-smmetric matrix of order 3 is zero. Statement-1 : For an matrix A, det(a T ) = det and det( A) = det. Where det(b) denotes the determinant of matrix B. Then : [AI-11] (1) Statement-1 is true and statement- is false () Both statements are true (3) Both statements are false (4) Statement-1 is false and statement- is true If 1 is the complex cube root of unit and matrix H, then H 7 is equal to: [AI-11] (1) H () (3) H (4) H 1 7. Let P and Q be 3 3 matrices with P Q. If P 3 = Q 3 and P Q = Q P, then determinant of (P + Q ) is equal to : [AI - 1 ] (1) 1 () (3) 1 (4) 63

15 1 8. Let 1 A 1. If u 1 and u are column matrices such that Au1 and Au 1, then u 1 + u is equal to : [AI-1] (1) () 1 1 (3) (4) 1 1 PRVIOUS YARS QUSTIONS ANSWR KY 64 XRCIS-5 [A] Que A ns Que A ns

16 XRCIS - 5 [B] J-[ADVANCD] : PRVIOUS YAR QUSTIONS a b c 1. If matrix A = b c a where a,b,c are real positive numbers, abc = 1 and A T A = I, then find the value of c a b a 3 + b 3 + c 3. [J 3, Mains M out of 6]. If A and A 3 = 15, then is equal to - [J 4 (Screening)] ±3 (B) ± ±5 3. If M is a 3 3 matrix, where M T M = I and det (M) = 1, then prove that det (M I) =. [J 4 (Mains), M out of 6] a 1 A 1 b d, 1 b c a 1 1 B d c, f g h f U g, h a x V, X = z If AX = U has infinitel man solutions, then prove that BX = V cannot have a unique solution. If further afd, then prove that BX = V has no solution [J 4 (Mains), 4M out of 6] 1 1 A 1, and A 1 (A ca d ), then the value of c and d are - 6, 11 (B) 6, 11 6, 11 6, 11 [J 5 (Screening)] 6. If 3 1 P, Comprehension (3 questions) A and Q = PAP T and x = P T Q 5 P, then x is equal to - (B) A 1, if U 1, U and U 3 are columns matrices satisfing. AU1, AU 3, 3 1 U is 3 3 matrix whose columns are U 1, U, U 3 then answer the following questions - (a) The value of U is - 3 (B) 3 3/ (b) The sum of the elements of U 1 is - 1 (B) 1 3 [J 5 (Screening)] AU and 65

17 (c) The value of 3 3 U is - [5] (B) [5/] [4] [3/] [J 6, 5 marks each] 8. Match the Statement / xpressions in Column I with the Statements / xpressions in Column II and indicate our answer b darkening the appropriate bubbles in the 4 4 matrix given in the ORS. Column I The minimum value of x x 4 x is (P) (B) Let A and B be 3 3 matrices of real numbers, (Q) 1 where A is smmetric, B is skew-smmetric, and (A+B)(A B) = (A B) (A + B). If (AB) t = ( 1) k AB, where (AB) t is the transpose of the matrix AB, then the possible values of k are Column II Let a = log 3 log 3. An integer k satisfing must be less than a ( k 3 ) 1, (R) If sin = cos, then the possible values of 1 are (S) 3 [J 8, 6] 9. Let A be the set of all 3 3 smmetric matrices all of whose entries are either or 1. Five of these entries are 1 and four of them are. (a) The number of matrices in A is - (b) (c) 1 (B) The number of matrices A in A for which the sstem of linear equations x 1 A z has a unique solution, is - less than 4 (B) at least 4 but less than 7 at least 7 but less than 1 at least 1 The number of matrices A in A for which the sstem of linear equations is inconsistent, is - x 1 A z (B) more than 1 1. (a) The number of 3 3 matrices A whose entries are either or 1 and for which the sstem has exactl two distinct solutions, is (B) [J 9, 4+4+4] x 1 A z

18 (b) Let k be a positive real number and let k 1 k k A k 1 k k k 1 and k 1 k B 1 k k. k k If det (adj A) + det(adj B) = 1 6, then [k] is equal to [Note : adj M denotes the adjoint of a square matrix M and [k] denotes the largest integer less than or equal to k]. ( c ) Let p be an odd prime number and T p be the following set of matrices : (i) a b Tp A : a,b,c,1,,...,p 1 c a The number of A in T p such that A is either smmetric or skew-smmetric or both, and det divisible b p is - (p 1) (B) (p 1) (p 1) + 1 p 1 (ii) The number of A in T p such that the trace of A is not divisible b p but det is divisible b p is - [Note : The trace of a matrix is the sum of its diagonal entries.] (p 1) (p p + 1) (B) p 3 (p 1) (p 1) (p 1) (p ) (iii) The number of A in T p such that det is not divisible b p is - p (B) p 3 5p p 3 3p p 3 p [J 1, ] 1 1. Let M and N be two 3 3 non-singular skew-smmetric matrices such that MN = NM. If P T denotes the transpose of P, then M N (M T N) 1 (MN 1 ) T is equal to - [J 11, 4] M (B) N M MN 1 a b 1. Let 1 be a cube root of unit and S be the set of all non-singular matrices of the form 1 c, 1 where each of a,b and c is either or. Then the number of distinct matrices in the set S is- (B) [J 11, 3, ( 1)] 1 3. Let M be 3 3 matrix satisfing 1 1 M 1, M 1 1 and M Then the sum of the diagonal entries of M is [J 11, 4] 1 4. Let P =[a ij ] be a 3 3 matrix and let Q = [b ij ], where b ij = i+j a ij for 1 < i, j < 3. If the determinant of P is, then the determinant of the matrix Q is - [J 1, 3M, 1M] 1 (B)

19 1 5. If P is a 3 3 matrix such that P T = P + I, where P T is the transpose of P and I is the 3 3 identit matrix, then there exists a column matrix x X z such that [J 1, 3M, 1M] PX (B) PX = X PX = X PX = X If the adjoint of a 3 3 matrix P is 1 7, then the possible value(s) of the determinant of P 1 3 is (are) - [J 1, 4M] (B) 1 1 PRVIOUS YARS QUSTIONS ANSWR KY A 5. C 6. A 7. (a) A, (b) B, (c) A 8. R (B) Q,S R,S P,R 9. (a) A, (b) B, (c) B 1. (a) A, (b) 4; (c) (i) D, (ii) C, (iii) D 1 1. Bonus 1. A D 1 5. D 1 6. A,D 68 XRCIS-5 [B]