Matrices and Determinants Advanced Level Pure Mathematics Advanced Level Pure Mathematics 8 Chapter 8 Matrices and Determinants 8.1 INTRODUCTION : MATRIX / MATRICES 2 8.2 SOME SPECIAL MATRIX 3 8.3 ARITHMETRICS OF MATRICES 4 8.4 INVERSE OF A SQUARE MATRIX 16 8.5 DETERMINANTS 19 8.6 PROPERTIES OF DETERMINANTS 21 8.7 INVERSE OF SQUARE MATRIX BY DETERMINANTS 27 Prepared by K. F. Ngai Page 1 Matrices and Determinants Advanced Level Pure Mathematics 8.1 INTRODUCTION : MATRIX / MATRICES 1. A rectangular array of mn numbers arranged in the form a11 a12 a1n a 21 a 22 a 2 n a m1 a m2 a mn is called an mn matrix. 2 3 4 e.g. is a 23 matrix. 1 8 5 2 e.g. 7 is a 31 matrix. 3 2. If a matrix has m rows and n columns, it is said to be order mn. 2 0 3 6 e.g. 3 4 7 0 is a matrix of order 34. 1 9 2 5 1 0 2 e.g. 2 1 5 is a matrix of order 3. 1 3 0 3. a 4. b1 b2 is called a column matrix or column vector. bn 1 a 2 a n is called a row matrix or row vector. 2 e.g. 7 is a column vector of order 31. 3 Prepared by K. F. Ngai Page 2 Matrices and Determinants e.g. 2 3 4 is a row vector of order 13. Advanced Level Pure Mathematics 5. If all elements are real, the matrix is called a real matrix. a11 a12 a1n a a 22 a 2 n 6. 21 is called a square matrix of order n. a n1 a n 2 a nn And a11 , a 22 , , a nn is called the principal diagonal. 3 9 e.g. is a square matrix of order 2. 0 2 7. Notation : a ij 8.2 m n a , ij m n , A , ... SOME SPECIAL MATRIX. Def.8.1 If all the elements are zero, the matrix is called a zero matrix or null matrix, denoted by Om n . 0 0 e.g. is a 22 zero matrix, and denoted by O2 . 0 0 Def.8.2 Let A a ij n n be a square matrix. (i) If aij 0 for all i, j, then A is called a zero matrix. (ii) If aij 0 for all i<j, then A is called a lower triangular matrix. (iii) If aij 0 for all i>j, then A is called a upper triangular matrix. a11 a 21 a n1 i.e. 0 a 22 an2 a11 a12 a1n 0 a 22 0 0 0 0 a nn Upper triangular matrix 0 0 a nn 0 0 Lower triangular matrix 1 0 0 e.g. 2 1 0 is a lower triangular matrix. 1 0 4 Prepared by K. F. Ngai Page 3 Matrices and Determinants Advanced Level Pure Mathematics 2 3 e.g. is an upper triangular matrix. 0 5 Def.8.3 Let A a ij n n be a square matrix. If aij 0 for all i j , then A is called a diagonal matrix. 1 0 0 e.g. 0 3 0 0 0 4 is a diagonal matrix. Def.8.4 If A is a diagonal matrix and a11 a 22 a nn 1 , then A is called an identity matrix or a unit matrix, denoted by I n . 1 0 0 I 3 0 1 0 0 0 1 e.g. 1 0 I2 , 0 1 8.3 ARITHMETRICS OF MATRICES. Def.8.5 Two matrices A and B are equal iff they are of the same order and their corresponding elements are equal. i.e. a ij m n bij m n e.g. a 2 1 c 4 b d 1 N.B. 2 3 2 4 4 0 3 0 Def.8.6 Let A a ij m n and aij bij for all i , j . a 1, b 1, c 2, d 4 . 2 1 3 0 2 3 1 1 4 1 0 4 and B bij m n . Define A B as the matrix C cij m n of the same order such that cij aij bij for all i=1,2,...,m and j=1,2,...,n. e.g. 2 3 1 2 4 3 1 0 4 2 1 5 Prepared by K. F. Ngai Page 4 Matrices and Determinants Advanced Level Pure Mathematics 2 1 2 3 1 N.B. 1. 3 0 is not defined. 1 4 1 0 4 2 3 5 is not defined. 2. 4 0 Def.8.7 Let A aij m n . Then A aij m n and A-B=A+(-B) e.g.1 1 2 3 2 4 0 B If A and 3 1 1 . Find -A and A-B. 1 0 2 Thm.8.1 Properties of Matrix Addition. Let A, B, C be matrices of the same order and O be the zero matrix of the same order. Then (a) A+B=B+A (b) (A+B)+C=A+(B+C) (c) A+(-A)=(-A)+A=O (d) A+O=O+A Def.8.8 Scalar Multiplication. Let A aij m n , k is scalar. Then kA is the matrix C cij by cij kaij , i.e. kA kaij e.g. defined i, j. m n 3 2 If A , 5 6 then N.B. m n -2A= ; 3 A 2 (1) -A=(-1)A (2) A-B=A+(-1)B Prepared by K. F. Ngai Page 5 Matrices and Determinants Advanced Level Pure Mathematics Thm.8.2 Properties of Scalar Multiplication. Let A, B be matrices of the same order and h, k be two scalars. Then (a) k(A+B)=kA+kB (b) (k+h)A=kA+hA (c) (hk)A=h(kA)=k(hA) Let A aij m n . The transpose of A, denoted by A T , or A , is Def.8.9 defined by a11 a A T 12 a1n e.g. 3 2 A , 5 6 e.g. 3 0 2 A , 4 6 1 e.g. A 5 , N.B. a 21 a m1 a 22 a m2 a 2 n a nm n m then A T then A T then A T (1) I T (2) A aij m n , then A T Thm.8.3 Properties of Transpose. Let A, B be two mn matrices and k be a scalar, then (a) ( A T ) T (b) ( A B ) T (c) (kA ) T Def.8.11 A square matrix A is called a symmetric matrix iff A T A . i.e. e.g. e.g. Def.8.12 A is symmetric matrix A T A aij a ji i, j 1 3 1 3 3 0 1 0 6 1 3 1 0 3 0 1 3 6 is a symmetric matrix. is not a symmetric matrix. A square matrix A is called a skew-symmetric matrix iff A T A . Prepared by K. F. Ngai Page 6 Matrices and Determinants Advanced Level Pure Mathematics i.e. A is skew-symmetric matrix A T A aij a ji i, j e.g.2 0 3 1 Prove that A 3 0 5 is a skew-symmetric matrix. 1 5 0 e.g.3 Is a ii 0 for all i=1,2,...,n for a skew-symmetric matrix? Def.12 Matrix Multiplication. Let A aik mn and B bkj matrix C cij m p n p . Then the product AB is defined as the mp where n cij ai 1b1 j ai 2 b2 j ain bnj aik bkj . k 1 i.e. n AB aik bkj k 1 m p Prepared by K. F. Ngai Page 7 Matrices and Determinants Advanced Level Pure Mathematics e.g.4 2 1 2 3 1 . Find AB and BA. Let A 3 0 and B 1 4 1 0 4 2 3 3 2 e.g.5 2 1 Let A 3 0 and B 1 4 3 2 N.B. Thm.8.4 1 0 . Find AB. Is BA well defined? 2 1 2 2 In general, AB BA . i.e. matrix multiplication is not commutative. Properties of Matrix Multiplication. (a) (AB)C = A(BC) (b) A(B+C) = AB+AC (c) (A+B)C = AC+BC (d) AO = OA = O (e) IA = AI = A (f) k(AB) = (kA)B = A(kB) Prepared by K. F. Ngai Page 8 Matrices and Determinants Advanced Level Pure Mathematics (g) ( AB) B A . T N.B. T T (1) Since AB BA ; Hence, A(B+C) (B+C)A and A(kB) (kB)A. (2) A 2 kA A( A kI ) ( A kI ) A . (3) AB AC O A(B C) O A O o rB C O 1 0 0 0 0 0 e.g. Let A , B , C 0 0 0 1 1 0 1 0 0 0 1 0 0 0 Then AB AC 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 But so Def. AO and B C, AB AC O A O or B C . Powers of matrices For any square matrix A and any positive integer n, the symbol A n denotes A A A A . n factors N.B. e.g.6 (1) ( A B ) 2 ( A B )( A B ) AA AB BA BB A 2 AB BA B 2 (2) If AB BA , then ( A B ) 2 A 2 2 AB B 2 2 1 1 1 2 3 2 4 0 Let A , B , C 1 0 and D 2 1 0 2 3 1 1 1 1 0 Evaluate the following : (a) ( A 2B)C (b) ( AC ) 2 (c) ( B T 3C ) T D (d) ( 2 A) T B DD T Prepared by K. F. Ngai Page 9 Matrices and Determinants Advanced Level Pure Mathematics Prepared by K. F. Ngai Page 10 Matrices and Determinants Advanced Level Pure Mathematics e.g.7 (a) Find a 2x2 matrix A such that 1 0 1 1 0 . 2 A 3 A 1 1 2 1 1 2 (b) Find a 2x2 matrix A 2 1 A T A and A 3 0 such that 2 1 . A 3 0 3 1 1 0 1 (c) If , find the values of x and . 1 1 x 0 x Prepared by K. F. Ngai Page 11 Matrices and Determinants Advanced Level Pure Mathematics e.g.8 e.g.9 cos sin Let A . Prove by mathematical induction that sin cos cos n sin n [HKAL92] (3 marks) An for n = 1,2, . sin n cos n (a) Let A a 1 where a , b R and a b . 0 b n Prove that A n a 0 a n bn a b for all positive integers n. bn 95 1 2 (b) Hence, or otherwise, evaluate . 0 3 [HKAL95] (6 marks) Prepared by K. F. Ngai Page 12 Matrices and Determinants Advanced Level Pure Mathematics e.g.10 0 1 0 (a) Let A 0 0 1 and B be a square matrix of order 3. Show that if A 0 0 0 and B are commutative, then B is a triangular matrix. (b) Let A be a square matrix of order 3. If for any x , y , z R , there exists x x R such that A y y , show that A is a diagonal matrix. z z (c) If A is a symmetric matrix of order 3 and A is nilpotent of order 2 (i.e. then A=O, where O is the zero matrix of order 3. A 2 O ), Prepared by K. F. Ngai Page 13 Matrices and Determinants Advanced Level Pure Mathematics Properties of power of matrices : (1) Let A be a square matrix, then ( A n ) T ( A T ) n . (2) If AB BA , then (a) ( A B) n A n C1n A n1 B C2n A n 2 B 2 C3n A n 3 B 3 Cnn1 AB n1 B n (b) ( AB ) n A n B n . (3) ( A I ) n A n C1n A n1 C2n A n 2 C3n A n 3 Cnn1 A Cnn I e.g.11 (a) Let X and Y be two square matrices such that XY = YX. (i) ( X Y ) 2 X 2 2 XY Y 2 Prove that n (ii) ( X Y ) n Crn X n r Y r for n = 3, 4, 5, ... . r0 (Note: For any square matrix A , define A 0 I .) (3 marks) 1 2 4 (b) By using (a)(ii) and considering 0 1 3 , or otherwise, find 0 0 1 1 2 4 0 1 3 0 0 1 100 . (4 marks) (c) If X and Y are square matrices, (i) prove that ( X Y ) 2 X 2 2 XY Y 2 implies XY = YX ; (ii) prove that ( X Y ) 3 X 3 3 X 2 Y 3 XY 2 Y 3 does NOT implies XY = YX . (Hint : Consider a particular X and Y, e.g. X 1 0 b 0 .) , Y 1 0 0 0 [HKAL90] (8 marks) Prepared by K. F. Ngai Page 14 Matrices and Determinants Advanced Level Pure Mathematics Prepared by K. F. Ngai Page 15 Matrices and Determinants Advanced Level Pure Mathematics 8.4 INVERSE OF A SQUARE MATRIX N.B. (1) If a, b, c are real numbers such that ab=c and b is non-zero, then c 1 cb 1 and b is usually called the multiplicative inverse of b. b C (2) If B, C are matrices, then is undefined. B a Def. A square matrix A of order n is said to be non-singular or invertible if and only if there exists a square matrix B such that AB = BA = I. The matrix B is called the multiplicative inverse of A, denoted by A 1 i.e. AA1 A1 A I . 3 5 2 5 . , show that the inverse of A is 1 2 1 3 e.g.12 Let A i.e. 3 5 1 2 1 2 5 . 1 3 2 5 1 3 e.g.13 Is 1 3 5 ? 1 2 Prepared by K. F. Ngai Page 16 Matrices and Determinants Advanced Level Pure Mathematics Def. If a square matrix A has an inverse, A is said to be non-singular or invertible. Otherwise, it is called singular or non-invertible. e.g. 3 5 and 2 5 are both non-singular. 1 2 1 3 i.e. A is non-singular iff A 1 exists. Thm. The inverse of a non-singular matrix is unique. N.B. (1) I 1 I , so I is always non-singular. (2) OA = O I , so O is always singular. (3) Since AB = I implies BA = I. Hence proof of either AB = I or BA = I is enough to assert that B is the inverse of A. e.g.14 Let A 2 1 . 7 4 (a) Show that I 6 A A 2 O . (b) Show that A is non-singular and find the inverse of A. (c) Find a matrix X such that AX 1 1 . 1 0 Prepared by K. F. Ngai Page 17 Matrices and Determinants Advanced Level Pure Mathematics Properties of Inverses Thm. Let A, B be two non-singular matrices of the same order and be a scalar. (a) ( A1 )1 A . (b) AT is a non-singular and ( AT )1 ( A1 )T . (c) A n is a non-singular and ( A n ) 1 ( A 1 ) n . (d) A is a non-singular and (A) 1 1 A 1 . (e) AB is a non-singular and ( AB) 1 B 1 A 1 . Proof Refer to Textbook P.228. Prepared by K. F. Ngai Page 18 Matrices and Determinants Advanced Level Pure Mathematics 8.5 DETERMINANTS Def. Let A aij be a square matrix of order n. The determinant of A, detA or |A| is defined as follows: a a12 (a) If n=2, det A 11 a11 a22 a12 a21 a21 a22 a11 a12 a13 (b) If n=3, det A a21 a22 a23 a 31 a32 a33 or det A a11 a22 a33 a21 a32 a13 a31 a12 a23 a31 a22 a13 a32 a23 a11 a33 a21 a12 e.g.15 Evaluate 1 3 (a) 4 1 3 1 2 (b) det 2 1 0 1 2 1 3 2 x e.g.16 If 8 x 1 0 , find the value(s) of x. 3 2 0 Prepared by K. F. Ngai Page 19 Matrices and Determinants Advanced Level Pure Mathematics N.B. a11 det A a 21 a 31 a11 or a12 a22 a32 a22 a32 a12 a21 a31 a13 a23 a33 a23 a a12 21 a33 a31 a23 a a22 11 a33 a31 a23 a a13 21 a33 a31 a13 a a32 11 a33 a21 a22 a32 a13 a23 or . . . . . . . . . By using e.g.17 Evaluate 3 2 0 (a) 0 1 1 0 2 3 0 2 0 (b) 8 2 1 3 2 3 Prepared by K. F. Ngai Page 20 Matrices and Determinants Advanced Level Pure Mathematics 8.6 PROPERTIES OF DETERMINANTS a1 (1) a2 a3 b1 b2 b3 c1 a1 c2 b1 c3 c1 a2 b2 c2 a3 b3 c3 i.e. det( A T ) det A . a1 b1 (2) a2 b2 a3 b3 c1 b1 c2 b2 c3 b3 a1 a2 a3 c1 b1 c2 b2 c3 b3 c1 c2 c3 a1 a2 a3 a1 a2 a3 c1 a2 c2 a1 c3 a3 b2 b1 b3 c2 a 2 c1 a3 c3 a1 b2 b3 b1 c2 c3 c1 b1 b2 b3 a1 (3) a2 a3 0 c1 a1 0 c2 0 a 2 0 c3 0 a1 (4) a2 a3 a1 a2 a3 c1 a1 c2 0 a1 c3 a3 b1 b2 0 c1 c2 0 b1 b1 b3 c1 c1 c3 a1 a1 a2 a3 (5) If , then a2 b1 b2 b3 a3 b1 b2 b3 a1 x1 (6) a2 x 2 a3 x 3 c1 x1 c2 x 2 c3 x 3 pa1 (7) pa2 pa3 pa1 pa2 pa3 N.B. b1 b2 b3 b1 b2 b3 pb1 pb2 pb3 c1 a1 c2 a 2 c3 a 3 c1 a1 c2 p a 2 c3 a3 b1 b2 b3 b1 b2 b3 pc1 a1 3 pc2 p a2 pc3 a3 pa1 (1) pa2 pa3 pb1 pb2 pb3 c1 c2 0 c3 c1 a1 c2 pa2 c3 a3 b1 b2 b3 b1 b2 b3 c1 c2 c3 b1 pb2 b3 c1 pc2 c3 c1 c2 c3 pc1 a1 pc2 p a2 pc3 a3 b1 b2 b3 c1 c2 c3 (2) If the order of A is n, then det(A) n det( A) Prepared by K. F. Ngai Page 21 Matrices and Determinants Advanced Level Pure Mathematics a1 (8) a2 a3 b1 b2 b3 c1 a1 b1 c2 a2 b2 c3 a3 b3 N.B. x1 x2 x3 y1 y2 y3 e.g.18 Evaluate b1 b2 b3 c1 c2 c3 z1 x1 y1 z1 C2 C3 C1 z2 x y 2 z 2 2 z3 x 3 y3 z3 1 2 0 (a) 0 4 5 , 6 7 8 y1 y2 y3 z1 z2 z3 5 3 7 (b) 3 7 5 7 2 6 1 a b c e.g.19 Evaluate 1 b c a 1 c ab Prepared by K. F. Ngai Page 22 Matrices and Determinants Advanced Level Pure Mathematics e.g.20 Factorize the determinant x y xy y xy x xy x y e.g.21 Factorize each of the following : a 3 b3 c 3 (a) a b c 1 1 1 [HKAL91] (4 marks) 2a 3 2b 3 2c 3 (b) a 2 b2 c2 1 a 3 1 b3 1 c 3 Prepared by K. F. Ngai Page 23 Matrices and Determinants Advanced Level Pure Mathematics Def. Multiplication of Determinants. a a12 b b Let A 11 , B 11 12 a 21 a 22 b21 b22 a11 a12 b11 b12 a 21 a 22 b21 b22 a b a12 b21 a11b12 a12 b22 11 11 a 21b11 a 22 b21 a 21b12 a 22 b22 Then A B Properties : (1) det(AB)=(detA)(detB) i.e. AB A B (2) |A|(|B||C|)=(|A||B|)|C| N.B. A(BC)=(AB)C (3) |A||B|=|B||A| N.B. ABBA in general (4) |A|(|B|+|C|)=|A||B|+|A||C| N.B. A(B+C)=AB+AC Prepared by K. F. Ngai Page 24 Matrices and Determinants Advanced Level Pure Mathematics 1 e.g.22 Prove that a a2 1 b b2 1 c (a b)(b c)(c a ) c2 Prepared by K. F. Ngai Page 25 Matrices and Determinants Advanced Level Pure Mathematics Minors and Cofactors a11 Let A a 21 a 31 Def. A11 Since a 22 a 32 a 23 a 33 A a 21 a13 a 23 , then Aij , the cofactor of aij , is defined by a 33 a a a a , A12 21 23 , ... , A33 11 12 . a 31 a 33 a 21 a 22 a12 a32 a12 a 22 a 32 a13 a33 + a 22 a11 a 31 a13 a a 23 11 a 33 a 31 a12 a 32 a21 A21 a22 A22 a23 A23 Thm. (a) det A if i j a i1 A j1 a i 2 A j 2 a i 3 A j 3 if i j 0 (b) det A if i j a1i A1 j a2i A2 j a3i A3 j if i j 0 e.g. a11 A11 a12 A12 a13 A13 det A , a11 A21 a12 A22 a13 A23 0 , etc. a11 a12 a 31 a 32 e.g.23 Let A a 21 a 22 a13 a 23 and cij be the cofactor of aij , where 1 i , j 3 . a 33 c11 (a) Prove that A c12 c13 c21 c22 c23 c11 (b) Hence, deduce that c12 c13 c31 c32 (det A) I c33 c21 c22 c23 c31 c32 (det A) 2 c33 Prepared by K. F. Ngai Page 26 Matrices and Determinants Advanced Level Pure Mathematics 8.7 INVERSE OF SQUARE MATRIX BY DETERMINANTS Def. A11 The cofactor matrix of A is defined as cofA A21 A31 Def. The adjoint matrix of A is defined as A11 adjA ( cofA) A12 A13 T A21 A22 A23 A12 A22 A32 A13 A23 . A33 A31 A32 . A33 e.g.24 If A a b , find adjA. c d e.g.25 1 1 3 (a) Let A 1 2 0 , find adjA. 1 1 1 3 2 1 (b) Let B 1 1 1 , find adjB. 5 1 1 Prepared by K. F. Ngai Page 27 Matrices and Determinants Advanced Level Pure Mathematics Thm. For any square matrix A of order n , A(adjA) = (adjA)A = (detA)I a11 a12 a1n A11 a a a A 2n A( adjA) 21 22 12 an1 an2 ann A1n A21 An1 A22 An2 A2 n Ann Prepared by K. F. Ngai Page 28 Matrices and Determinants Advanced Level Pure Mathematics Thm. Let A be a square matrix. If detA 0 , then A is non-singular and A1 1 adjA . det A Proof Let the order of A be n , from the above theorem , e.g.26 3 2 1 Given that A 1 1 1 , find A 1 . 5 1 1 1 AadjA I det A e.g.27 Suppose that the matrix A a b 1 is non-singular , find A . c d Prepared by K. F. Ngai Page 29 Matrices and Determinants Advanced Level Pure Mathematics e.g.28 Given that A 3 5 , 1 2 Thm. find A 1 . A square matrix A is non-singular iff detA 0 . e.g.29 Show that A 3 5 is non-singular. 1 2 Prepared by K. F. Ngai Page 30 Matrices and Determinants Advanced Level Pure Mathematics e.g.30 x 1 2 x 1 Let A x 1 2 1 , where x R . 7 x 5 (a) Find the value(s) of x such that A is non-singular. (b) If x=3 , find A 1 . N.B. A is singular (non-invertible) iff A 1 does not exist. Thm. A square matrix A is singular iff detA = 0. Prepared by K. F. Ngai Page 31 Matrices and Determinants Advanced Level Pure Mathematics Properties of Inverse matrix. Let A, B be two non-singular matrices of the same order and be a scalar. (1) (A) 1 1 A 1 (2) ( A 1 ) 1 A (3) ( A T ) 1 A 1 (4) ( A n ) 1 A 1 T n for any positive integer n. (5) ( AB ) 1 B 1 A 1 (6) The inverse of a matrix is unique. (7) det( A 1 ) N.B. 1 det A XY 0 X 0 or Y 0 (8) If A is non-singular , then AX 0 A1 AX A0 0 X 0 N.B. XY XZ X 0 or Y Z (9) If A is non-singular , then AX AY A1 AX A1 AY X Y (10) ( A 1 MA) n ( A 1 MA)( A 1 MA) ( A 1 MA) a 1 a 0 0 (11) If M 0 b 0 , then M 1 0 0 0 c 0 0 b 1 0 A 1 M n A 0 0. c 1 an 0 0 a 0 0 (12) If M 0 b 0 , then M n 0 b n 0 where n 0 . n 0 0 c 0 0 c 4 1 0 1 3 1 1 0 0 e.g.31 Let A 1 3 1 , B 0 13 and 4 M 0 1 0 . 0 3 1 0 33 10 0 0 2 (a) Find A 1 and M 5 . (b) Show that ABA 1 M . (c) Hence, evaluate B 5 . Prepared by K. F. Ngai Page 32 Matrices and Determinants Advanced Level Pure Mathematics 3 8 2 4 . and P 1 1 1 5 e.g.32 Let A (a) Find P 1 AP . (b) Find A n , where n is a positive integer. [HKAL94] (6 marks) Prepared by K. F. Ngai Page 33 Matrices and Determinants Advanced Level Pure Mathematics e.g.33 (a) Show that if A is a 3x3 matrix such that A A , then detA=0. t 1 2 74 (b) Given that B 2 1 67 , 74 67 1 use (a) , or otherwise , to show det(I B) 0 . Hence deduce that det( I B 4 ) 0 . [HKAL93] (7 marks) e.g.34 (a) If , and are the roots of x 3 px q 0 , find a cubic equation whose roots are 2 , 2 and 2 . x 2 3 (b) Solve the equation 2 x 3 0 . 2 3 x Hence, or otherwise, solve the equation x 3 38x 2 361x 900 0 . [HKAL94] (6 marks) Prepared by K. F. Ngai Page 34 Matrices and Determinants Advanced Level Pure Mathematics e.g.35 Let M be the set of all 2x2 matrices. For any A a11 a21 a12 M , a22 define tr ( A) a11 a22 . (a) Show that for any A, B, C M and , R, (i) tr (A B ) tr ( A) tr ( B ) , (ii) tr ( AB) tr (BA) , (iii) the equality “ tr ( ABC) tr (BAC) ” is not necessary true. (5 marks) (b) Let A M. (i) Show that A 2 tr ( A) A (det A) I , where I is the 2x2 identity matrix. (ii) If tr ( A 2 ) 0 and tr ( A) 0 , use (a) and (b)(i) to show that A is singular and A 2 0 . (5 marks) (c) Let S, T M such that (ST TS )S S (ST TS ) . Using (a) and (b) or otherwise, show that ( ST TS ) 2 0 [HKAL92] (5 marks) Prepared by K. F. Ngai Page 35 Matrices and Determinants Advanced Level Pure Mathematics Prepared by K. F. Ngai Page 36 Matrices and Determinants Advanced Level Pure Mathematics e.g.36 Eigenvalue and Eigenvector 3 1 and let x denote a 2x1 matrix. 2 0 Let A (a) Find the two real values 1 and 2 of with 1 > 2 such that the matrix equation (*) Ax x has non-zero solutions. (b) Let x1 and x2 be non-zero solutions of (*) corresponding to 1 and 2 respectively. Show that if x x1 11 x21 x x2 12 x22 and then the matrix X x11 x21 x12 is non-singular. x22 (c) Using (a) and (b), show that and hence 1n A X 0 n 0 AX X 1 0 2 0 1 where n is a positive integer. X 2n 3 1 Evaluate . 2 0 n [HKAL82] Prepared by K. F. Ngai Page 37