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11.1Matrices 11.2Determinants 11.3Inverses of Square Matrices Chapter Summary Case Study Matrices and Determinants 11.

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Presentation on theme: "11.1Matrices 11.2Determinants 11.3Inverses of Square Matrices Chapter Summary Case Study Matrices and Determinants 11."— Presentation transcript:

1 11.1Matrices 11.2Determinants 11.3Inverses of Square Matrices Chapter Summary Case Study Matrices and Determinants 11

2 P. 2  Team X produce 500 pieces of product A, 200 pieces of product B and 350 pieces of product C  Team Y produce 200 pieces of product A, 400 pieces of product B and 450 pieces of product C Case Study Contents of  product A:1.5 kg of copper, 0.2 kg of steel  product B:0.6 kg of copper, 1.4 kg of steel  product C:0.8 kg of copper, 1 kg of steel How to organize and calculate the total amount of copper and steel needed by each team? Great! Please calculate the total amount of materials needed by each team. We received an order to produce three kinds of products. Teams X and Y will work together to finish this job. (1) Amount of copper needed by Team X ? (2) Amount of steel needed by Team X ? (3) Amount of copper needed by Team Y ? (4) Amount of steel needed by Team Y ? That’s tedious!

3 P. 3 Organization We can arrange the data in tabular form: Calculation (1)Amount of copper needed by Team X  Product AProduct BProduct C Team X Team Y Copper (in kg)Steel (in kg) Product A Product B Product C0.81 CopperSteel Team X1150 kg Team Y (500  1.5  200  0.6  350  0.8) kg  1150 kg (2)Amount of steel needed by Team X ? (3)Amount of copper needed by Team Y ? (4)Amount of steel needed by Team Y ? 730 kg 900 kg1050 kg 1st row 1st column 2nd column Case Study

4 P. 4 A rectangular array of numbers arranged in m rows and n columns is called a m  n matrix Matrices An m  n matrix is represented in the form or A matrix with m rows and n columns is said to be a matrix of order m  n. The number a ij in the ith row and the jth column of a matrix is called an element or entry. For example, in the 2  3 matrix, a 12  4 and a 23  7. 3rd column 2nd row mth row nth column A. Introduction

5 P. 5 For an a m  n matrix, if m  1, it has only 1 row and is called a row matrix; if n  1, it has only 1 column and is called a column matrix. We should specify the row number first, then the column number. ( ) is a row matrix of order 1  3. is a column matrix of order 3  1. Two matrices are said to be equal if they satisfy the following definition: Equality of Matrices Two matrices A  (a ij ) m  n and B  (b ij ) m  n are equal if and only if they have the same order and their corresponding elements are equal, i.e., a ij  b ij for all i = 1, 2, 3,..., m and j = 1, 2, 3,..., n Matrices A. Introduction

6 P. 6 Example 11.1T Solution: If, find the values of w, x, y and z. From the definition, w  2, x  4, y  6 and z  Matrices A. Introduction

7 P. 7 Zero Matrix A zero matrix, or a null matrix, is a matrix that all its elements are zero. For example, is a 2  3 matrix. Square Matrix A square matrix is a matrix with the same numbers of rows and columns. For example, is a square matrix of order 2. Notes: The order of a square matrix is denoted by its number of rows n Matrices B. Special Types of Matrices

8 P. 8 For example, is the identity matrix of order 3. Identity Matrix An identity matrix of order n, which is denoted by I, is an n  n square matrix with. An identity matrix is also called a unit matrix Matrices B. Special Types of Matrices

9 P. 9 Some rules on the operations of matrices: Addition of Matrices Suppose A  (a ij ) m  n and B  (b ij ) m  n are two matrices of order m  n. Then the sum of A and B is also an m  n matrix C  (c ij ) m  n with c ij  a ij  b ij, for all i  1, 2, 3,..., m and j  1, 2, 3,..., n. For example, if and, then Note that the addition of matrices is defined only when the two matrices are of the same order Matrices C. Operations of Matrices

10 P. 10 Negative of Matrices Let A  (a ij ) m  n be an m  n matrix. The negative of A, denoted by  A, is the matrix whose elements are the negative of the corresponding elements of A, i.e.,  A  (  a ij ) m  n, for all i  1, 2, 3,..., m and j  1, 2, 3,..., n. For example, if, then. Subtraction of Matrices Suppose A  (a ij ) m  n and B  (b ij ) m  n are two matrices of order m  n. The difference of A and B is defined as A  B  A  (  B) Matrices C. Operations of Matrices

11 P. 11 Example 11.2T Solution: Suppose and. Find the matrix Z such that Y  Z  X. ∵ Y  Z  X ∴ Z  X  Y When summing up matrices, we sum up each pair of the corresponding elements independently Matrices C. Operations of Matrices

12 P. 12 Properties of Matrix Addition Let A  (a ij ) m  n, B  (b ij ) m  n and C  (c ij ) m  n be m  n matrices and 0 be the m  n zero matrix. Then we have: (a)A  B  B  A(Commutative Law) (b)(A  B)  C  A  (B  C)(Associative Law) (c)A  0  0  A  A (d)A  (  A)  (  A)  A  0 Proofs of (a) and (b): By the definition of addition of matrices, A  B  (a ij ) m  n  (b ij ) m  n  (a ij  b ij ) m  n  (b ij  a ij ) m  n  (b ij ) m  n  (a ij ) m  n  B  A B  A (A  B)  C  (a ij  b ij ) m  n  (c ij ) m  n  [(a ij  b ij )  c ij ] m  n  [a ij  (b ij  c ij )] m  n  (a ij ) m  n  (b ij  c ij ) m  n  A  (B  C) 11.1 Matrices C. Operations of Matrices

13 P. 13 Scalar Multiplication of Matrices The scalar multiplication of an m  n matrix A  (a ij ) m  n and a real number k, which is denoted by kA, is an m  n matrix whose elements are the corresponding elements of A multiplied by k, i.e., kA  (ka ij ) m  n, for all i  1, 2, 3,..., m and j  1, 2, 3,..., n. For example,. Properties of Scalar Multiplication Let A and B be two m  n matrices and h, k be two real numbers. We have (a)k(A  B)  kA  kB;(Distributive Law) (b)(h  k)A  hA  kA; (c)hkA  h(kA)  k(hA) Matrices C. Operations of Matrices

14 P. 14 Example 11.3T Solution: Suppose and. Evaluate 2X  3Y and 4Y  2X. 2X  3Y4Y  2X 11.1 Matrices C. Operations of Matrices

15 P. 15 Multiplication of Matrices Let A  (a ij ) m  n be an m  n matrix and B  (b ij ) n  p be an n  p matrix. The product AB is an m  p matrix C  (c ij ) m  p where c ij  a i1 b 1j  a i2 b 2j ...  a in b nj , for all i  1, 2, 3, …, m and j  1, 2, 3, …, p. To understand the process of the multiplication of matrices, students may also refer to the Case Study at the beginning of this chapter. Notes: When calculating the product AB, the matrix A should be placed on the left while B is placed on the right. Multiplication of matrices is non-commutative, i.e., for two matrices A and B, AB  BA in general Matrices C. Operations of Matrices

16 P. 16 Suppose and. ∴ AB is a 2  2 matrix. Also consider the product BA. ∵ B is a 3  2 matrix and A is a 2  3 matrix. ∴ BA is a 3  3 matrix. ∴ AB  BA ∵ A is a 2  3 matrix and B is a 3  2 matrix Matrices C. Operations of Matrices

17 P. 17 Example 11.4T Solution: For each of the following pairs of matrices X and Y, find XY and YX. (a), (b), (a)XY 11.1 Matrices C. Operations of Matrices

18 P. 18 Example 11.4T Solution: For each of the following pairs of matrices X and Y, find XY and YX. (a), (b), (a) YX 11.1 Matrices C. Operations of Matrices

19 P. 19 Example 11.4T Solution: For each of the following pairs of matrices X and Y, find XY and YX. (a), (b), (b)XY YX is undefined. The number of columns of Y is not equal to the number of rows of X Matrices C. Operations of Matrices

20 P. 20 Even though A  0 and B  0, we still have AB  0: Suppose, and. ∴ AB  0 does not imply A  0 or B  0. ∴ AB  AC does not imply A  0 or B  C  0. Consider AB  AC The following shows AC  0: ∵ A  0 and B  C. AB  AC  0 A(B  C)  Matrices C. Operations of Matrices

21 P. 21 Example 11.5T Solution: Let. Find a non-zero square matrix B of order 2 such that (a)AB  0,(b)BA  0. Let, where a, b, c and d are some constants. (a) ∴ c  d  0 ∵ AB  0 ∴ (b) ∴ b  d  0 ∵ BA  0 ∴ 11.1 Matrices C. Operations of Matrices

22 P. 22 Properties of Matrix Multiplication Let h and k be real numbers and A, B and C be matrices such that the following matrix products are defined. We have: (a)(AB)C  A(BC); (Associative Law) (b)(i)A(B + C)  AB + AC; (ii)(A + B)C  AC + BC; (c)k(AB)  (kA)B  A(kB); (d)(hA)(kB)  (hk)AB; (e)A0  0A  0, where A is a square matrix and 0 is a zero square matrix; (f)AI  IA  A, where A is a square matrix and I is an identity matrix. (Distributive Law) Remarks: The proofs are left for students Matrices C. Operations of Matrices

23 P. 23 Power of Square Matrices For any square matrix A and any positive integer n, we have For square matrices A and B of same order: 1.(A  B) 2  (A  B)(A  B)  AA  AB  BA  BB  A 2  AB  BA  B 2 2.(A  B)(A  B)  AA  AB  BA  BB  A 2  AB  BA  B 2 In general, (A  B) 2  A 2  2AB  B 2 and (A  B)(A  B)  A 2  B 2. The expressions cannot be reduced to the form we learnt in junior form unless AB  BA Matrices C. Operations of Matrices

24 P. 24 Example 11.6T Solution: Let. (a)X 2 X 2  XX. X 2 is also a 3  3 matrix. (a)Find the matrix X 2. (b)Hence, find the matrix 3X 2  2X  4I, where I is the 3  3 identity matrix Matrices C. Operations of Matrices

25 P. 25 Example 11.6T Solution: Let. (b) 3X 2  2X  4I (a)Find the matrix X 2. (b)Hence, find the matrix 3X 2  2X  4I, where I is the 3  3 identity matrix Matrices C. Operations of Matrices

26 P. 26 Example 11.7T Solution: If, show, by mathematical induction, that for all positive integers n. For n  1, obviously L.H.S.  R.H.S. ∴ The proposition is true for n  1. Assume the proposition is true for some positive integers k, that is,. When n  k  1, L.H.S.  X k  1  R.H.S. ∴ The proposition is true for n  k  1. The following shows an outline of solution only. Students should show your workings clearly. When n  k  1, show that R.H.S Matrices C. Operations of Matrices

27 P. 27 Transpose of Matrix Let A  (a ij ) m  n be an m  n matrix. The transpose of matrix of A, denoted by A t or A T, is an n  m matrix A t  (c ij ) n  m such that c ij  a ji for all i  1, 2, … n and j  1, 2, …, m. The transpose of a matrix A is obtained by interchanging the rows and the columns in A, for examples: 11.1 Matrices C. Operations of Matrices

28 P. 28 Properties of Transposes Let A and B be two m  n matrices, we have (a)(A t ) t  A; (b)(A  B) t  A t  B t ; (c)(kA) t  kA t, where k is any constant. Let A be an m  n matrix and B be an n  p matrix, we have (d)(AB) t  B t A t. Remarks: The proofs are left for students Matrices C. Operations of Matrices

29 P. 29 Example 11.8T Given that and. If (A t ) 2  pA t  qI  0, find the values of p and q. Solution: ∵ ( A t ) 2  pA t  qI  0 ∴ By comparing the corresponding elements of the matrices on both sides, we have p  9 and q  Matrices C. Operations of Matrices

30 P. 30 For an n  n square matrix A, denote the determinant of A by. Similar to matrices, only determinants of at most order 3 will de discussed. Determinant of Order 2 For a 2  2 square matrix A , the value of its determinant, which is denoted by | A| or det A, is defined by  a 11 a 22  a 12 a 21. a 11 a 22  a 12 a 21 is called the expansion of the determinant Determinants A. Introduction

31 P. 31 Example 11.9T If  5, find the value of x. Solution: A. Introduction 11.2 Determinants

32 P. 32 To memorize the expansion of the determinant:  This rule is only applicable for determinants of order 3. Notes: This rule is called the rule of Sarrus. Determinant of Order 3 For a 3  3 square matrix A , the value of its determinant is defined by  a 11 a 22 a 33  a 12 a 23 a 31  a 13 a 21 a 32  a 13 a 22 a 31  a 11 a 23 a 32  a 12 a 21 a Determinants A. Introduction

33 P. 33 Example 11.10T Solution: Evaluate the following determinants. (a)(b) (a)  (  2)(  2)(1)  1(  1)(  4)  3(5)(0)  3(  2)(  4)  (  2)(  1)(0)  1(5)(1)   21 (b)  a(1)(0)  0(b)(1)  1(0)(c)  1(1)(1)  a(b)(c)  0(0)(0)   (1  abc) 11.2 Determinants A. Introduction

34 P. 34 Example 11.11T Solution: Let a, b, c, d and e be five distinct numbers. If, prove that c(ae  bd)  a  e  b  d Determinants A. Introduction

35 P. 35 The following shows some of the properties of determinants, which are true for determinants of any order. For any square matrix A, the determinant of A is equal to that of the transpose of A, i.e.,. If any two rows (or columns) of a matrix are interchanged, the determinant changes sign but its absolute value remains unchanged. e.g., ;. These properties can be verified by expanding of the determinants. Remarks: B. Properties of Determinants 11.2 Determinants

36 P. 36 If all the elements in any one row (or column) of a matrix are multiplied by a factor, then the determinant is just the product of the original determinant and the factor. e.g., for any k. For example, if, then (i), (ii). B. Properties of Determinants 11.2 Determinants

37 P. 37 The determinant of a matrix is zero if all the elements in a row (or column) are zero, i.e.,. When k  0, we have: When all the elements are also multiplied by k, we have: If all the elements of an n  n square matrix are multiplied by the same factor, then the resulting determinant is the product of the original determinant and the nth power of the factor, i.e.,. B. Properties of Determinants 11.2 Determinants

38 P. 38 The determinant of a matrix is zero if the elements of a row (or a column) are proportional to those of another row (or another column), i.e., if. In particular, we have: If any two rows (or columns) of a matrix are equal, the determinant is equal to zero, i.e.,. B. Properties of Determinants 11.2 Determinants

39 P. 39 Consider the result of addition of matrices, we have: If all the elements in any row (or column) of a matrix can be expressed as the sum of two terms, then the determinant can also be expressed as the sum of the two determinants, i.e.,. When p, q and r are proportional to the elements of the other row, we have: If all the elements in a row (or column) of a matrix is added or subtracted by multiples of the other row (or column), then the value of the determinant will remain unchanged, i.e., for any k. B. Properties of Determinants 11.2 Determinants

40 P. 40 Finally, for the product of two square matrices, we have: For any n  n square matrices A and B, the product of their determinants is equal to the determinant of the matrix AB, i.e.,. Verification: Let and. Then. L.H.S.R.H.S. B. Properties of Determinants 11.2 Determinants

41 P. 41 Example 11.12T Solution: Without expanding the determinant, show that. Take out the common factor 7 from C 2 R1  R2  R1;R3  R2  R3R1  R2  R1;R3  R2  R3 In order to show the value of the determinant equals to zero, we need to show that any two rows or columns are the same. B. Properties of Determinants 11.2 Determinants

42 P. 42 Example 11.13T Solution: Without expanding the determinant, show that 3 is a factor of. (Given that the determinant is non-zero.) C2  C3  C2C2  C3  C2 Take out the common factor 3 from C 2 Since all the elements in the determinant are integers, its value in an integer. ∴ 3 is a factor of the given determinant. B. Properties of Determinants 11.2 Determinants

43 P. 43 Consider the determinant. The expansion of the determinant  aei  bfg  cdh  ceg  afh  bdi  a(ei  fh)  b( fg  di)  c(dh  eg) Group the a terms, the b terms and the c terms; Arrange in alphabetical order  a(ei  fh)  b(di  fg)  c(dh  eg) The  /  sign of each term is determined by the position of a, b, c as shown below: C. Evaluation of Determinants of Order Determinants

44 P. 44 The expansion of the determinant  aei  bfg  cdh  ceg  afh  bdi  b( fg  di)  e(ai  cg)  h(cd  af ) Group the b terms, the e terms and the h terms; Arrange in alphabetical order  b(di  fg)  e(ai  cg)  h(af  cd) The  /  sign of each term is determined by the position of b, e, h as shown below: Consider the determinant. C. Evaluation of Determinants of Order Determinants

45 P. 45 Summarize the results as follows: The determinant of order 3 can be expanded along any row or column, i.e., or, etc. For each of the element,  minor corresponding determinant obtained  cofactor product of the minor and the sign of the term Remarks: In a determinant of order 3: For each of the elements a, c, e, g and i, cofactor  minor; For each of the elements b, d, f and h, cofactor   (minor). C. Evaluation of Determinants of Order Determinants

46 P. 46 Example 11.14T Evaluate the determinant by expanding along (a)the first row,(b)the third column. Solution: (a)Value of the determinant   1(  6  48)  7(15  72)  4(  30  18)  615 (b)Value of the determinant  4(  30  18)  8(  6  63)  3(  2  35)  615 C. Evaluation of Determinants of Order Determinants

47 P. 47 Example 11.15T Solution: Show that. Hence evaluate. R3  R1  R3R3  R1  R3 ∴ Expand along R 3 C. Evaluation of Determinants of Order Determinants

48 P. 48 Example 11.16T Let and. Find the determinant of AB. Solution: ∴ Students may try to find the matrix AB first, and then the determinant of AB. However, each of the determinants of A and B can be evaluated more easily in this case. C. Evaluation of Determinants of Order Determinants

49 P. 49 Example 11.17T Factorize. Solution: C. Evaluation of Determinants of Order Determinants

50 P. 50 Example 11.18T Prove that. Solution: C1  C2  C3  C1C1  C2  C3  C1 ∴ C. Evaluation of Determinants of Order Determinants

51 P. 51 Example 11.19T Solve the equation. Solution:, where x  0. When x  0, the determinant becomes. R1  R3  R1R1  R3  R1 C. Evaluation of Determinants of Order Determinants

52 P. 52 For a non-zero real number n, is a multiplicative inverse of n such that  n  1, i.e., n  1 n  1. For matrices, matrix division is not defined. We can try to find a matrix B such that BA  AB  I. Inverse of a Matrix If square matrices A and B of order n satisfy the relationship AB = BA = I, where I is the identity matrix of order n, then the matrix B is called the inverse of A and denoted by A  1, i.e., AA  1  A  1 A  I Inverses of Square Matrices A. Introduction

53 P. 53 For example, consider and ∴ B is the inverse of A and A is the inverse of B. In particular, the inverse of an identity matrix is the identity matrix itself. Note the following relation in real numbers: 1  1  1  Inverses of Square Matrices A. Introduction

54 P. 54 Actually, not all square matrices have their corresponding inverses. Singular and Non-singular Matrices A square matrix A is said to be non-singular or invertible if and only if its inverse exists. Otherwise, it is said to be singular or non-invertible. If the inverse of a square matrix exists, then we have: Uniqueness of Inverse The inverse of a non-singular square matrix is unique. Proof (using contraction): Suppose B and C are two distinct inverse matrices of A, i.e.,AB  BA  I and AC  CA  I. ThenB  BI  BAC  IC  C, which contradicts to B  C Inverses of Square Matrices A. Introduction

55 P. 55 Consider the matrix. Suppose, i.e.,. By comparing the corresponding elements of the matrices on both sides, we have: determinant of A transpose of cofactors of A For a 2  2 square matrix, the cofactors of a, b, c and d are d,  c,  b and a respectively Inverses of Square Matrices A. Introduction

56 P. 56 For example, if, then Inverse of a 2  2 matrix Let. If A is non-singular, then the inverse of A is given by: The determinant of A is Inverses of Square Matrices A. Introduction

57 P. 57 Inverse of a 3  3 matrix Let. If A is non-singular, then the inverse of A is given by: cofactor transpose of cofactors of A 11.3 Inverses of Square Matrices A. Introduction

58 P. 58 In general, the inverse of a non-singular matrix A contains the factor. Thus if, the matrix is singular. Theorem A square matrix A is non-singular if an only if. Proof: ‘if ’:If A is non-singular, then there exists a matrix B such that AB  BA  I. ∵, and ∴ ‘only if ’:If, then we can find the inverse: ∴ A is non-singular Inverses of Square Matrices A. Introduction

59 P. 59 Example 11.20T Find the inverses of the following matrices. (a)(b) Solution: (a) 11.3 Inverses of Square Matrices A. Introduction

60 P. 60 Example 11.20T Find the inverses of the following matrices. (a)(b) Solution: (b) 11.3 Inverses of Square Matrices A. Introduction

61 P. 61 Example 11.21T Let P be a square matrix such that 2I  P  P 2  0. Prove that P is non-singular and find P  1 in terms of P and I. Solution: 2I  P  P 2  0 P  P 2  2I P(I  P)  2I ∴ P is non-singular and Inverses of Square Matrices A. Introduction

62 P. 62 Properties of Inverses Let A and B be two non-singular square matrices of the same order, k be a non-zero real number, and n be a positive integer. Then (a)(A  1 )  1  A; (b)(kA)  1  k  1 A  1 ; (c)(A n )  1  (A  1 ) n ; (d)(A t )  1  (A  1 ) t ; (e) ; (f)(AB)  1  B  1 A  1. Proof of (f): ∵ (AB)(B  1 A  1 )  A(BB  1 )A  1  AIA  1  AA  1  I and (B  1 A  1 )(AB)  B  1 (A  1 A)B  B  1 IB  B  1 B  I ∴ By definition, (AB)  1  B  1 A  Inverses of Square Matrices B. Properties of Inverses

63 P. 63 Example 11.22T Let and. (a)Find A  1 and B  1. (b)Hence find (AB 2 )  1 and [(AB) t ]  1. Solution: (a) 11.3 Inverses of Square Matrices B. Properties of Inverses

64 P. 64 Example 11.22T Let and. (a)Find A  1 and B  1. (b)Hence find (AB 2 )  1 and [(AB) t ]  1. Solution: (b) (AB 2 )  1  (B 2 )  1 A  1  (B  1 ) 2 A  Inverses of Square Matrices B. Properties of Inverses

65 P. 65 Example 11.22T Let and. (a)Find A  1 and B  1. (b)Hence find (AB 2 )  1 and [(AB) t ]  1. Solution: [(AB) t ]  1  [(AB)  1 ] t  (B  1 A  1 ) t (b) 11.3 Inverses of Square Matrices B. Properties of Inverses

66 P. 66 Example 11.23T Let. (a)Find M 2. (b)Hence find M  1. Solution: (a) (b) 11.3 Inverses of Square Matrices B. Properties of Inverses

67 P. 67 Example 11.24T Let. (a)Find X  1. (b)Hence find Y if YX . Solution: (a) 11.3 Inverses of Square Matrices B. Properties of Inverses

68 P. 68 Example 11.24T (a)Find X  1. (b)Hence find Y if YX . Solution: Let. (b)Y  (YX)X  Inverses of Square Matrices B. Properties of Inverses

69 P. 69 Example 11.25T Let and. Solution: (a)Find the matrix Y  1 XY. (b)Hence find X The following shows an outline of solution only. Students should show your workings clearly. (a) 11.3 Inverses of Square Matrices B. Properties of Inverses

70 P. 70 Example 11.25T Let and. Solution: (a)Find the matrix Y  1 XY. (b)Hence find X (b)Consider (Y  1 XY) 1000  (Y  1 XY)(Y  1 XY)(Y  1 …) … (… Y)(Y  1 XY)  Y  1 X(I) X(I) … (I) XY  Y  1 X 1000 Y ∴ Y(Y  1 XY) 1000 Y  1  X 1000 ∵ ∴ 11.3 Inverses of Square Matrices B. Properties of Inverses

71 P Matrices Chapter Summary 1.Definition An m  n matrix is represented in the form An m  n matrix may also be represented by the symbol (a ij ) m  n or [a ij ] m  n.

72 P Matrices 2.Operations of Matrices Let A  (a ij ) m  n and B  (b ij ) m  n be two matrices and k be a real number. (a)Addition A  B  (a ij  b ij ) m  n, for all i  1, 2,..., m and j  1, 2,..., n (b)Subtraction A  B  (a ij  (  1)b ij ) m  n, for all i  1, 2,..., m and j  1, 2,..., n (c)Scalar Multiplication kA  (ka ij ) m  n (d)Transpose A t  (c ij ) n  m where c ij  a ji, for all i  1, 2,..., n and j  1, 2,..., m (e)Multiplication Let A  (a ij ) m  n, B  (b ij ) n  p and C  (c ij ) m  p. If AB  C, then c ij  a i1 b 1j  a i2 b 2j ...  a in b nj  Chapter Summary

73 P Determinant of order 2  a 11 a 22  a 12 a 21 2.Determinant of order 3   a 11 a 22 a 33  a 12 a 23 a 31  a 13 a 21 a 32  a 13 a 22 a 31  a 11 a 23 a 32  a 12 a 21 a Determinants Chapter Summary

74 P Definition For a square matrix A, if there exists a matrix B such that AB  BA  I, then B is called the inverse of A and is denoted by A  1. 2.Inverse of a 2  2 matrix: 3.Inverse of a 3  3 matrix: 11.3 Inverses of Square Matrices Chapter Summary

75 Follow-up 11.1 If ( )  ( i j k ), find the values of i, j and k. Solution: From the definition, i  3, j  5 and k  Matrices A. Introduction

76 Follow-up 11.2 Solution: Suppose and. Find the matrix C such that A  B  C. ∵ A  B  C ∴ C  A  B 11.1 Matrices C. Operations of Matrices

77 Follow-up 11.3 Solution: Suppose and. Evaluate 2P  Q and P  4Q. 2P  Q2P  QP  4Q 11.1 Matrices C. Operations of Matrices

78 Solution: Follow-up 11.4 For each of the following pairs of matrices P and Q, find PQ and QP. (a), (b), (a)PQ 11.1 Matrices C. Operations of Matrices

79 Solution: Follow-up 11.4 For each of the following pairs of matrices P and Q, find PQ and QP. (a), (b), (a) QP In general, for two square matrices P and Q of the same order, PQ  QP Matrices C. Operations of Matrices

80 Follow-up 11.4 Solution: For each of the following pairs of matrices P and Q, find PQ and QP. (a), (b), (b)PQ QP is undefined. The number of columns of Q is not equal to the number of rows of P Matrices C. Operations of Matrices

81 Follow-up 11.5 Solution: Let. Find a non-zero square matrix B of order 2 such that (a)AB  0,(b)BA  0. Let, where a, b, c and d are some constants. (a) ∴ a  b  0 ∵ AB  0 ∴ (b) ∴ a  b  0and c  d  0 ∵ BA  0 ∴ b   aand d   c 11.1 Matrices C. Operations of Matrices

82 Solution: Follow-up 11.6 Let. (a)Find the matrices A 2 and A 4. (b)Hence find the matrix A 4  2A 2  3I, where I is the 2  2 identity matrix. (a)A 2 A 4  A 2 A Matrices C. Operations of Matrices

83 Follow-up 11.6 Solution: Let. (a)Find the matrices A 2 and A 4. (b)Hence find the matrix A 4  2A 2  3I, where I is the 2  2 identity matrix. (b)A 4  2A 2  3I 11.1 Matrices C. Operations of Matrices

84 Follow-up 11.7 Solution: If, show, by mathematical induction, that for all positive integers n. For n  1, ∴ The proposition is true for n  1. L.H.S. R.H.S. Assume the proposition is true for some positive integers k, that is, Matrices C. Operations of Matrices

85 Follow-up 11.7 Solution: If, show, by mathematical induction, that for all positive integers n. When n  k  1, L.H.S.  M k  1 ∴ The proposition is true for n  k  1. ∴ By the principle of mathematical induction, the proposition is true for all positive integers n Matrices C. Operations of Matrices

86 Follow-up 11.8 Given that and. If (A t ) 2  pA t  qI  0, find the values of p and q. Solution: ∵ ( A t ) 2  pA t  qI  0 ∴ By comparing the corresponding elements of the matrices on both sides, we have p   6 and q  Matrices C. Operations of Matrices

87 Follow-up 11.9 If  9, find the possible values of x. Solution: or A. Introduction 11.2 Determinants

88 Follow-up Solution: Evaluate the following determinants. (a)(b)  9  1  a 2  b 2  c 2 (a)  (  1)(  3)(  5)  0(0)(4)  2(0)(0)  2(  3)(4)  (  1)(0)(0)  0(0)(  5) (b)  1(1)(1)  a(b)(c)  (  c)(  a)(  b)  (  c)(1)(c)  1(b)(  b)  a(  a)(1) A. Introduction 11.2 Determinants

89 Follow-up Solution: Let a, b, c and d be four distinct numbers. If, prove that (a  d)(c  b)  0. A. Introduction 11.2 Determinants

90 Follow-up Solution: Without expanding the determinant, show that. Take out the common factor  3 from C 2 C3  C2  C3C3  C2  C3 In order to show the value of the determinant equals to zero, we need to show that any two rows or columns are the same. B. Properties of Determinants 11.2 Determinants

91 Follow-up Solution: Without expanding the determinant, show that is divisible by 25. C1  C3  C1C1  C3  C1 C1  C2  C1C1  C2  C1 Take out the common factor 5 from C 1 B. Properties of Determinants 11.2 Determinants

92 Follow-up Solution: Without expanding the determinant, show that is divisible by 25. C3  C1  C3C3  C1  C3 ∴ The given determinant is divisible by 25. integer B. Properties of Determinants 11.2 Determinants

93 Follow-up Evaluate the determinant by expanding along (a)the third column,(b)the second row. Solution: (a)Value of the determinant   2(63  12)  8(9  24)  5(3  42)   45 (b)Value of the determinant   7(  30  18)  3(5  8)  8(9  24)   45 C. Evaluation of Determinants of Order Determinants

94 Follow-up Solution: Expand along R 3 Show that. Hence evaluate. R 3  R 1  R 3 ; R 2  4  R 1  R 2 ∴ C. Evaluation of Determinants of Order Determinants

95 Follow-up Let and. Find the determinant of AB. Solution: ∴ C. Evaluation of Determinants of Order Determinants

96 Follow-up Factorize. Solution: R 2  R 1  R 2 ; R 3  R 1  R 3 c 2  (a  b) 2  (c  a  b)(c  a  b) C 1  C 2  C 3  C 1 (b  c) 2  a 2  (b  c  a)(b  c  a) C. Evaluation of Determinants of Order Determinants

97 Follow-up Factorize. Solution: C. Evaluation of Determinants of Order Determinants

98 Follow-up Prove that. Solution: R1  R2  R1;R2  R3  R2R1  R2  R1;R2  R3  R2 Expand (a  b  c)(c  a) on the R.H.S. to complete the proof. Take out the common factors from R 1 and R 2 C. Evaluation of Determinants of Order Determinants

99 Solution: Follow-up Solve the equation. or R1  R3  R1;R2  R3  R2R1  R3  R1;R2  R3  R2 Take out the common factors from R 1 and R 2 C. Evaluation of Determinants of Order Determinants

100 Follow-up Find the inverses of the following matrices. (a)(b) Solution: (a) 11.3 Inverses of Square Matrices A. Introduction

101 Follow-up Solution: Find the inverses of the following matrices. (a)(b) (b) 11.3 Inverses of Square Matrices A. Introduction

102 Follow-up Let X be a square matrix such that 2X 2  4X  5I  0. Prove that X is non-singular and find X  1 in terms of X and I. Solution: 2X 2  4X  5I  0 2X 2  4X  5I 2X (X  2I )  5I ∴ X is non-singular and Inverses of Square Matrices A. Introduction

103 Follow-up Let and. (a)Find M  1 and N  1. (b)Hence find (M 2 N)  1 and (NM t )  1. Solution: (a) 11.3 Inverses of Square Matrices B. Properties of Inverses

104 Follow-up Solution: (b) (M 2 N)  1  N  1 (M 2 )  1  N  1 (M  1 ) 2 Let and. (a)Find M  1 and N  1. (b)Hence find (M 2 N)  1 and (NM t )  Inverses of Square Matrices B. Properties of Inverses

105 Follow-up Solution: (NM t )  1  (M t )  1 N  1  (M  1 ) t N  1 (b) Let and. (a)Find M  1 and N  1. (b)Hence find (M 2 N)  1 and (NM t )  Inverses of Square Matrices B. Properties of Inverses

106 Follow-up Let. (a)Find P 2. (b)Hence find P  1. Solution: (a) (b) 11.3 Inverses of Square Matrices B. Properties of Inverses

107 Follow-up Let. (a)Find P  1. (b)Hence find Q if PQ . Solution: (a) 11.3 Inverses of Square Matrices B. Properties of Inverses

108 Follow-up Let. (a)Find P  1. (b)Hence find Q if PQ . Solution: (b)Q  P  1 (PQ) 11.3 Inverses of Square Matrices B. Properties of Inverses

109 Follow-up Let and. (a)Show that.(b)Hence find Q 800. Solution: (a) 11.3 Inverses of Square Matrices B. Properties of Inverses

110 Follow-up Solution: Let and. (a)Show that.(b)Hence find Q 800. (b) Consider (P  1 QP) 800  (P  1 QP)(P  1 QP)(P  1 …) … (… P)(P  1 QP)  P  1 Q(I) Q(I) … (I) QP  P  1 Q 800 P ∴ P(P  1 QP) 800 P  1  Q 800 ∵ ∴ 11.3 Inverses of Square Matrices B. Properties of Inverses


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