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Matrices and Determinants 2nd Class – Mechanical Eng. Dep. Mathematics4 Matrices and Determinants INTRODUCTION : MATRIX / MATRICES 2 SOME SPECIAL MATRIX 3 ARITHMETRICS OF MATRICES 4 INVERSE OF A SQUARE MATRIX 16 DETERMINANTS 19 PROPERTIES OF DETERMINANTS 21 INVERSE OF SQUARE MATRIX BY DETERMINANTS 27 Prepared by Eng. Baseem Adnan Al-twajre Page 1 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 a2 a n is called a row matrix or row vector. 2 e.g. 7 is a column vector of order 31. 3 Prepared by Eng. Baseem Adnan Al-twajre Page 2 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. e.g. 2 3 4 is a row vector of order 13. 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 , a22 , , ann is called the principal diagonal. 3 9 e.g. is a square matrix of order 2. 0 2 7. Notation : aij m n a , ij m n , A , ... SOME SPECIAL MATRIX. Def.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..2 Let A aij 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 Eng. Baseem Adnan Al-twajre Page 3 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 2 3 e.g. is an upper triangular matrix. 0 5 Def..3 Let A aij 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..4 If A is a diagonal matrix and a11 a22 ann 1 , then A is called an identity matrix or a unit matrix, denoted by I n . 1 0 , I2 0 1 e.g. 1 0 0 I 3 0 1 0 0 0 1 ARITHMETRICS OF MATRICES. Def..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 e.g. a 2 1 c 4 b d 1 N.B. 2 3 2 4 4 0 3 0 Def..6 Let A aij m n m n aij bij for all i , j . and 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 Eng. Baseem Adnan Al-twajre Page 4 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 2 3 1 2 4 1 2 3 1 is not defined. N.B. 1. 0 1 0 4 4 3 2. 5 is not defined. 0 Def..7 Let A aij mn . Then A aij mn and A-B=A+(-B) e.g.1 1 2 3 2 4 0 . Find -A and A-B. If A and B 3 1 1 1 0 2 Thm..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 Scalar Multiplication. Let A aij mn , k is scalar. Then kA is the matrix C cij cij kaij , i, j . i.e. kA kaij e.g. m n defined by m n 3 2 If A , 5 6 then N.B. -2A= ; 3 A 2 (1) -A=(-1)A (2) A-B=A+(-1)B Prepared by Eng. Baseem Adnan Al-twajre Page 5 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. Thm..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) Def..9 Let A aij mn . The transpose of A, denoted by A T , or A , is 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. Thm..3 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 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..11 A square matrix A is called a symmetric matrix iff A T A . i.e. e.g. e.g. 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. Def..12 A square matrix A is called a skew-symmetric matrix iff A T A . Prepared by Eng. Baseem Adnan Al-twajre Page 6 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 aii 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 Eng. Baseem Adnan Al-twajre Page 7 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 0 . Find AB. Is BA well defined? 2 1 2 2 1 4 3 2 N.B. Thm..4 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 Eng. Baseem Adnan Al-twajre Page 8 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. (g) ( AB) T B T A T . N.B. (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 2 B )C (b) ( AC ) 2 (c) ( B T 3C ) T D (d) (2A) T B DD T Prepared by Eng. Baseem Adnan Al-twajre Page 9 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. Prepared by Eng. Baseem Adnan Al-twajre Page 10 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 such that 2 1 2 1 . A T A and A A 3 0 3 0 3 1 1 0 1 (c) If , find the values of x and . 1 1 x 0 x Prepared by Eng. Baseem Adnan Al-twajre Page 11 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. e.g.8 e.g.9 cos sin Let A . Prove by mathematical induction that sin cos cos n sin n 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 a 0 n a n bn a b for all positive integers n. n b 95 (b) Hence, or otherwise, evaluate 1 2 . 0 3 Prepared by Eng. Baseem Adnan Al-twajre Page 12 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 0 1 0 e.g.10 (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. A2 O ), Prepared by Eng. Baseem Adnan Al-twajre Page 13 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 .) 1 2 4 (b) By using (a)(ii) and considering 0 1 3 , or otherwise, find 0 0 1 100 1 2 4 0 1 3 . 0 0 1 (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 Prepared by Eng. Baseem Adnan Al-twajre Page 14 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. Prepared by Eng. Baseem Adnan Al-twajre Page 15 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 cb 1 and b 1 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. AA 1 A1 A I . 5 . 1 3 e.g.12 Let A 3 5 , show that the inverse of A is 2 1 2 1 2 5 . 1 3 i.e. 3 5 1 2 e.g.13 Is 2 5 1 3 1 3 5 ? 1 2 Prepared by Eng. Baseem Adnan Al-twajre Page 16 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 if 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 6A A2 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 Eng. Baseem Adnan Al-twajre Page 17 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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) An is a non-singular and ( An ) 1 ( A1 ) n . (d) A is a non-singular and (A) 1 1 A 1 . (e) AB is a non-singular and ( AB) 1 B 1 A1 . Prepared by Eng. Baseem Adnan Al-twajre Page 18 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 a31 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 1 2 3 (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 Eng. Baseem Adnan Al-twajre Page 19 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. N.B. a11 det A a21 a31 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 Eng. Baseem Adnan Al-twajre Page 20 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. PROPERTIES OF DETERMINANTS a1 (1) a2 a3 b1 b2 b3 c1 a1 c2 b1 c3 c1 a2 b2 c2 a1 (2) a2 a3 b1 b2 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 b1 b2 b3 c1 a2 c2 a1 c3 a3 b2 b1 b3 c2 a 2 c1 a3 c3 a1 b2 b3 b1 c2 c3 c1 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 a3 b3 c3 b1 b2 0 b1 b1 b3 i.e. det( AT ) det A . c1 c2 0 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 Eng. Baseem Adnan Al-twajre Page 21 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 y2 z2 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 Eng. Baseem Adnan Al-twajre Page 22 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 2a 3 2b 3 2c 3 (b) a 2 b2 c2 1 a 3 1 b3 1 c 3 Prepared by Eng. Baseem Adnan Al-twajre Page 23 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. Def. Multiplication of Determinants. a a12 b b Let A 11 , B 11 12 a21 a22 b21 b22 a11 a12 b11 b12 a21 a22 b21 b22 a b a b a11b12 a12 b22 11 11 12 21 a21b11 a22 b21 a21b12 a22 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 Eng. Baseem Adnan Al-twajre Page 24 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 1 e.g.22 Prove that a a2 1 b b2 1 c (a b)(b c)(c a ) c2 Prepared by Eng. Baseem Adnan Al-twajre Page 25 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. Minors and Cofactors a11 a13 a 23 , then Aij , the cofactor of a ij , is defined by a 31 a 32 a 33 a 23 , A12 a 21 a 23 , ... , A33 a11 a12 . a 31 a 33 a 33 a 21 a 22 a12 Let A a 21 a 22 Def. A11 Since a 22 a 32 A a 21 a12 a13 a32 a33 + a 22 a11 a 31 a13 a a 23 11 a 33 a 31 a12 a 32 a21A21 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. e.g.23 a11 Let A a 21 a 31 a12 a 22 a 32 a13 a 23 and cij be the cofactor of a ij , 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 Eng. Baseem Adnan Al-twajre Page 26 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 1 1 3 0 , find adjA. 1 1 1 e.g.25 (a) Let A 1 2 3 2 1 (b) Let B 1 1 1 , find adjB. 5 1 1 Prepared by Eng. Baseem Adnan Al-twajre Page 27 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. Thm. For any square matrix A of order n , A(adjA) = (adjA)A = (detA)I a11 a A( adjA) 21 an1 a12 a1n A11 a22 a2 n A12 an2 ann A1n A21 An1 A22 An2 A2 n Ann Prepared by Eng. Baseem Adnan Al-twajre Page 28 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 Eng. Baseem Adnan Al-twajre Page 29 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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 Eng. Baseem Adnan Al-twajre Page 30 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. x 1 2 x 1 1 , where x R . 7 x 5 e.g.30 Let A x 1 2 (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) if A 1 does not exist. Thm. A square matrix A is singular iff detA = 0. Prepared by Eng. Baseem Adnan Al-twajre Page 31 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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) (11) a 1 a 0 0 If M 0 b 0 , then M 1 0 0 0 c 0 (12) e.g.31 0 b 1 0 A 1 M n A 0 0. c 1 an 0 0 a 0 0 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 Let A 1 3 1 , B 0 13 and M 0 1 0 . 4 0 3 1 0 33 10 0 0 2 (a) Find A 1 and M 5 . (b) Show that ABA1 M . (c) Hence, evaluate B 5 . Prepared by Eng. Baseem Adnan Al-twajre Page 32 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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. Prepared by Eng. Baseem Adnan Al-twajre Page 33 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. e.g.33 (a) Show that if A is a 3x3 matrix such that A t A , then detA=0. 2 74 1 67 , 74 67 1 1 (b) Given that B 2 use (a) , or otherwise , to show det(I B) 0 . Hence deduce that det( I B 4 ) 0 . 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 . Prepared by Eng. Baseem Adnan Al-twajre Page 34 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. 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. (b) Let A M. (i) Show that A2 tr ( A) A (det A) I , where I is the 2x2 identity matrix. (ii) If tr ( A2 ) 0 and tr ( A) 0 , use (a) and (b)(i) to show that A is singular and A2 0 . (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 Prepared by Eng. Baseem Adnan Al-twajre Page 35 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. Prepared by Eng. Baseem Adnan Al-twajre Page 36 Matrices and Determinants 2nd Class – Mechanical Eng. Dep. e.g.36 Eigenvalue and Eigenvector Let A 3 1 and let x denote a 2x1 matrix. 2 0 (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 and x x2 12 x22 then the matrix X x11 x12 is non-singular. x21 x22 (c) Using (a) and (b), show that 1n A X 0 and hence n 0 AX X 1 0 2 0 1 X where n is a positive integer. 2n Evaluate 3 1 . n 2 0 Prepared by Eng. Baseem Adnan Al-twajre Page 37