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11.1 Matrices

11. Matrices and Determinants. Case Study. 11.1 Matrices. 11.2 Determinants. 11.3 Inverses of Square Matrices. Chapter Summary. We received an order to produce three kinds of products. Teams X and Y will work together to finish this job.

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11.1 Matrices

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

  2. We received an order to produce three kinds of products. Teams X and Y will work together to finish this job. Great! Please calculate the total amount of materials needed by each team. (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 ? Case Study  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 That’s tedious! 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?

  3. 1st column 1st column 2nd column 2nd column 1st row 1st row Case Study Organization We can arrange the data in tabular form: Calculation (1) Amount of copper needed by Team X  (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 ?

  4. 3rd column nth column 2nd row mth row For example, in the 2  3 matrix , a12  4 and a23  7. 11.1 Matrices A. Introduction A rectangular array of numbers arranged in m rows and n columns is called a m  nmatrix. An m  n matrix is represented in the form or Amatrixwithmrowsandncolumnsissaidtobeamatrixofordermn. The number aij in the ith row and the jth column of a matrix is called an element or entry.

  5. We should specify the row number first, then the column number. is a column matrix of order 3  1. 11.1 Matrices A. Introduction 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. (5 4 3) is a row matrix of order 1  3. Two matrices are said to be equal if they satisfy the following definition: Equality of Matrices Two matrices A  (aij)m  n and B  (bij)m  n are equal if and only if they have the same order and their corresponding elements are equal, i.e., aij  bij for all i = 1, 2, 3, ... , m and j = 1, 2, 3, ... , n.

  6. If , find the values of w, x, y and z. From the definition, w  2, x  4, y  6 and z  10. 11.1 Matrices A. Introduction Example 11.1T Solution:

  7. For example, is a 2  3 matrix. For example, is a square matrix of order 2. 11.1 Matrices B. Special Types of Matrices Zero Matrix A zero matrix, or a null matrix, is a matrix that all its elements are zero. Square Matrix A square matrix is a matrix with the same numbers of rows and columns. Notes: The order of a square matrix is denoted by its number of rows n.

  8. 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. For example, is the identity matrix of order 3. 11.1 Matrices B. Special Types of Matrices

  9. For example, if and , then Note that the addition of matrices is defined only when the two matrices are of the same order. 11.1 Matrices C. Operations of Matrices Some rules on the operations of matrices: Addition of Matrices Suppose A  (aij)m  n and B  (bij)m  nare two matrices of order m  n. Then the sum of A and B is also an m  n matrix C (cij)m  n with cij  aij  bij, for all i  1, 2, 3, ... , m and j  1, 2, 3, ... , n.

  10. For example, if , then . 11.1 Matrices C. Operations of Matrices Negative of Matrices Let A  (aij)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(aij)m  n, for all i 1, 2, 3, ... , m and j 1, 2, 3, ... , n. Subtraction of Matrices Suppose A  (aij)m  n and B  (bij)m  nare two matrices of order m  n.The difference of A and B is defined as A  B  A  (B).

  11. Suppose and . Find the matrix Z such that Y  Z  X. When summing up matrices, we sum up each pair of the corresponding elements independently. 11.1 Matrices C. Operations of Matrices Example 11.2T Solution: ∵ YZ  X ∴ Z  X  Y

  12. 11.1 Matrices C. Operations of Matrices Properties of Matrix Addition Let A (aij)m  n, B (bij)m  n and C (cij)m  nbe 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  (aij)m  n  (bij)m  n (A  B) C  (aij  bij)m  n  (cij)m  n  (aij  bij)m  n  [(aij  bij)  cij]m  n  (bij  aij)m  n  [aij  (bij  cij)]m  n  (bij)m  n  (aij)m  n  (aij)m  n  (bij  cij)m  n B  A A  (B  C)

  13. For example, . 11.1 Matrices C. Operations of Matrices Scalar Multiplication of Matrices The scalar multiplication of an m  n matrix A (aij)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 (kaij)m  n, for all i 1, 2, 3, ... , m and j 1, 2, 3, ... , n. 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).

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

  15. Multiplication of Matrices Let A (aij)m  n be an m  n matrix and B (bij)n  p be an npmatrix.TheproductABisanmpmatrixC(cij)m  p where cijai1b1j  ai2b2j  ...  ainbnj , 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. 11.1 Matrices C. Operations of Matrices 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.

  16. Suppose and . 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 11.1 Matrices C. Operations of Matrices ∵ A is a 2 3 matrix and B is a 3 2 matrix. ∴ AB is a 2  2 matrix.

  17. 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 Example 11.4T Solution:

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

  19. For each of the following pairs of matrices X and Y, find XY and YX. (a) , (b) , (b) XY The number of columns of Y is not equal to the number of rows of X. 11.1 Matrices C. Operations of Matrices Example 11.4T Solution: YX is undefined.

  20. Suppose , and . The following shows AC  0: 11.1 Matrices C. Operations of Matrices Even though A  0 and B  0, we still have AB  0: ∴ AB  0 does not imply A  0 or B  0. Consider AB  AC AB  AC  0 A(B  C) 0 ∵ A  0 and B  C. ∴ AB  AC does not imply A  0 or B  C  0.

  21. 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) (b) ∴ ∴ 11.1 Matrices C. Operations of Matrices Example 11.5T Solution: ∵ AB  0 ∵ BA  0 ∴ c  d  0 ∴ b  d  0

  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) 11.1 Matrices C. Operations of Matrices Remarks: The proofs are left for students.

  23. Power of Square Matrices For any square matrix A and any positive integer n, we have The expressions cannot be reduced to the form we learnt in junior form unless AB  BA. 11.1 Matrices C. Operations of Matrices For square matrices A and B of same order: 1. (A  B)2 (A  B)(A  B) AA  AB  BA  BB A2  AB  BA  B2 2. (A  B)(A  B) AA  AB  BA  BB A2  AB  BA  B2 In general, (A  B)2  A2  2AB  B2 and (A  B)(A  B)  A2  B2.

  24. Let . (a) X2 X2  XX. X2 is also a 3  3 matrix. 11.1 Matrices C. Operations of Matrices Example 11.6T (a) Find the matrix X2. (b) Hence, find the matrix 3X2  2X  4I, where I is the 3  3 identity matrix. Solution:

  25. Let . (b) 3X2  2X  4I 11.1 Matrices C. Operations of Matrices Example 11.6T (a) Find the matrix X2. (b) Hence, find the matrix 3X2  2X  4I, where I is the 3  3 identity matrix. Solution:

  26. If , show, by mathematical induction, that for all positive integers n. The following shows an outline of solution only. Students should show your workings clearly. Assumethepropositionistrueforsomepositive integers k, that is, . When n k1, show that R.H.S. . 11.1 Matrices C. Operations of Matrices Example 11.7T Solution: For n  1, obviously L.H.S.  R.H.S. ∴ The proposition is true for n  1. When n  k  1, L.H.S.  Xk  1  R.H.S. ∴ The proposition is true for n  k  1.

  27. 11.1 Matrices C. Operations of Matrices Transpose of Matrix Let A (aij)m  n be an m  n matrix. The transpose of matrix of A, denoted by Ator AT, is an n  m matrix At (cij)n  m such that cij aji 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:

  28. 11.1 Matrices C. Operations of Matrices Properties of Transposes Let A and B be two m  n matrices, we have (a) (At)tA; (b) (A  B)tAt  Bt; (c) (kA)tkAt, where k is any constant. Let A be an m  n matrix and B be an n  p matrix, we have (d) (AB)tBtAt. Remarks: The proofs are left for students.

  29. Given that and . If (At)2  pAt  qI  0, find the values of p and q. ∴ By comparing the corresponding elements of the matrices on both sides, we have p  9 and q  8 . 11.1 Matrices C. Operations of Matrices Example 11.8T Solution: ∵ (At )2  pAt  qI  0

  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 a11a22  a12a21. a11a22  a12a21 is called the expansion of the determinant. 11.2 Determinants A. Introduction

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

  32. Determinant of Order 3 For a 3  3 square matrix A , the value of its determinant is defined by a11a22a33  a12a23a31  a13a21a32  a13a22a31  a11a23a32  a12a21a33. This rule is called the rule of Sarrus. 11.2 Determinants A. Introduction To memorize the expansion of the determinant:      Notes: This rule is only applicable for determinants of order 3.

  33. 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 Example 11.10T Solution:

  34. Let a, b, c, d and e be five distinct numbers. If , prove that c(ae  bd)  a  e  b  d. 11.2 Determinants A. Introduction Example 11.11T Solution:

  35. 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., ; . 11.2 Determinants B. Properties of Determinants The following shows some of the properties of determinants, which are true for determinants of any order. Remarks: These properties can be verified by expanding of the determinants.

  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) . 11.2 Determinants B. Properties of Determinants

  37. The determinant of a matrix is zero if all the elements in a row (or column) are zero, i.e., . 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., . 11.2 Determinants B. Properties of Determinants When k  0, we have: When all the elements are also multiplied by k, we have:

  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 . If any two rows (or columns) of a matrix are equal, the determinant is equal to zero, i.e., . 11.2 Determinants B. Properties of Determinants In particular, we have:

  39. 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., . 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. 11.2 Determinants B. Properties of Determinants Consider the result of addition of matrices, we have: When p, qand r areproportional to theelementsof theother row,wehave:

  40. 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 . 11.2 Determinants B. Properties of Determinants Finally, for the product of two square matrices, we have: L.H.S. R.H.S.

  41. Without expanding the determinant, show that . Take out the common factor 7 from C2 R1  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. 11.2 Determinants B. Properties of Determinants Example 11.12T Solution:

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

  43. Consider the determinant . Group the a terms, the b terms and the c terms; Arrange in alphabetical order The / sign of each term is determined by the position of a, b, c as shown below: 11.2 Determinants C. Evaluation of Determinants of Order 3 The expansion of the determinant  aei  bfg  cdh  ceg  afh  bdi  a(ei  fh)  b(fg  di)  c(dh  eg)  a(ei  fh)  b(di  fg)  c(dh  eg)

  44. Consider the determinant . Group the b terms, the e terms and the h terms; Arrange in alphabetical order The / sign of each term is determined by the position of b, e, h as shown below: 11.2 Determinants C. Evaluation of Determinants of Order 3 The expansion of the determinant  aei  bfg  cdh  ceg  afh  bdi  b(fg  di)  e(ai  cg) h(cd  af)  b(di  fg)  e(ai  cg) h(af  cd)

  45. The determinant of order 3 can be expanded along any row or column, i.e., or , etc. 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). 11.2 Determinants C. Evaluation of Determinants of Order 3 Summarize the results as follows: Remarks: For each of the element,  minor corresponding determinant obtained  cofactor product of the minor and the sign of the term

  46. Evaluate the determinant by expanding along (a) the first row, (b) the third column. (a) Value of the determinant  615  615 (b) Value of the determinant 11.2 Determinants C. Evaluation of Determinants of Order 3 Example 11.14T Solution:  1(6  48)  7(15  72)  4(30  18)  4(30  18)  8(6  63)  3(2  35)

  47. Show that . Hence evaluate . R3  R1  R3 ∴ Expand along R3 11.2 Determinants C. Evaluation of Determinants of Order 3 Example 11.15T Solution:

  48. Let and . Find the determinant of AB. 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. ∴ 11.2 Determinants C. Evaluation of Determinants of Order 3 Example 11.16T Solution:

  49. Factorize . 11.2 Determinants C. Evaluation of Determinants of Order 3 Example 11.17T Solution:

  50. Prove that . C1  C2  C3  C1 ∴ 11.2 Determinants C. Evaluation of Determinants of Order 3 Example 11.18T Solution:

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