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# 4.5 matrix notation

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### 4.5 matrix notation

1. 1. Matrix Notation
2. 2. Matrix NotationA matrix is a rectangular table of numbers.
3. 3. Matrix NotationA matrix is a rectangular table of numbers.For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4are matrices.
4. 4. Matrix NotationA matrix is a rectangular table of numbers.For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4are matrices. Systems of linear equations can be put intomatrices then solved using matrix notation and eliminationmethod.
5. 5. Matrix NotationA matrix is a rectangular table of numbers.For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4are matrices. Systems of linear equations can be put intomatrices then solved using matrix notation and eliminationmethod.For example, if we put the following system into a matrix, x + 4y = –7 Eq. 1{2x – 3y = 8 Eq. 2
6. 6. Matrix NotationA matrix is a rectangular table of numbers.For example -1 2 -1 5 2 -3 6 -2 3 4 -1 0 11 9 -4are matrices. Systems of linear equations can be put intomatrices then solved using matrix notation and eliminationmethod.For example, if we put the following system into a matrix, x + 4y = –7 Eq. 1{2x – 3y = 8 Eq. 2we get 1 4 -7 2 -3 8
7. 7. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers.
8. 8. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.
9. 9. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.The Three Row Operations for Matrices
10. 10. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.The Three Row Operations for MatricesI. Switching rows, notated as Row i Row j, or Ri Rj
11. 11. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.The Three Row Operations for MatricesI. Switching rows, notated as Row i Row j, or Ri RjFor example,1 4 -72 -3 8
12. 12. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.The Three Row Operations for MatricesI. Switching rows, notated as Row i Row j, or Ri RjFor example,1 4 -7 R1 R2 2 -3 82 -3 8 1 4 -7
13. 13. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.The Three Row Operations for MatricesI. Switching rows, notated as Row i Row j, or Ri RjFor example,1 4 -7 R1 R2 2 -3 82 -3 8 1 4 -7II. Multiply Row i by a nonzero constant k, notated as k*Ri.
14. 14. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.The Three Row Operations for MatricesI. Switching rows, notated as Row i Row j, or Ri RjFor example,1 4 -7 R1 R2 2 -3 82 -3 8 1 4 -7II. Multiply Row i by a nonzero constant k, notated as k*Ri.For example:1 4 -72 -3 8
15. 15. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.The Three Row Operations for MatricesI. Switching rows, notated as Row i Row j, or Ri RjFor example,1 4 -7 R1 R2 2 -3 82 -3 8 1 4 -7II. Multiply Row i by a nonzero constant k, notated as k*Ri.For example:1 4 -7 -3*R22 -3 8
16. 16. Matrix NotationEach row of the matrix corresponds to an equation andeach column corresponds to a variable except that the lastcolumn corresponds to numbers. Operations of theequations correspond to operations of rows in the matrices.There’re three main operations.The Three Row Operations for MatricesI. Switching rows, notated as Row i Row j, or Ri RjFor example,1 4 -7 R1 R2 2 -3 82 -3 8 1 4 -7II. Multiply Row i by a nonzero constant k, notated as k*Ri.For example: 1 4 -7 1 4 -7 -3*R2 2 -3 8 -6 9 -24
17. 17. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.
18. 18. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, 1 4 -7 2 -3 8
19. 19. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, 1 4 -7 -2*R1 add  R2 2 -3 8
20. 20. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 2 -3 8
21. 21. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22
22. 22. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22Fact: Performing row operations on a matrix does not changethe solution of the system.
23. 23. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22Fact: Performing row operations on a matrix does not changethe solution of the system.Elimination Method in Matrix Notation
24. 24. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22Fact: Performing row operations on a matrix does not changethe solution of the system.Elimination Method in Matrix Notation• Apply row operations to transform the matrix to the upperdiagonal form where all the entries below the main diagonal(the lower left triangular region) are 0.
25. 25. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22Fact: Performing row operations on a matrix does not changethe solution of the system.Elimination Method in Matrix Notation• Apply row operations to transform the matrix to the upperdiagonal form where all the entries below the main diagonal(the lower left triangular region) are 0.* * * ** * * ** * * *
26. 26. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22Fact: Performing row operations on a matrix does not changethe solution of the system.Elimination Method in Matrix Notation• Apply row operations to transform the matrix to the upperdiagonal form where all the entries below the main diagonal(the lower left triangular region) are 0.* * * * Row operations* * * ** * * *
27. 27. Matrix NotationIII. Add the multiple of row on top of another row, notated as “k*Ri add  Rj”.For example, write a copy of -2*R1 -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22Fact: Performing row operations on a matrix does not changethe solution of the system.Elimination Method in Matrix Notation• Apply row operations to transform the matrix to the upperdiagonal form where all the entries below the main diagonal(the lower left triangular region) are 0.* * * * Row operations * * * ** * * * 0 0 * * * 0* * * * * *
28. 28. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable.
29. 29. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable.
30. 30. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.
31. 31. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2
32. 32. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix:
33. 33. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: 1 4 -7 2 -3 8
34. 34. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: 1 4 -7 2 -3 8
35. 35. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: 1 4 -7 -2*R1 add  R2 2 -3 8
36. 36. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 2 -3 8
37. 37. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22
38. 38. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22From the bottom R2: -11y = 22  y = -2
39. 39. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22From the bottom R2: -11y = 22  y = -2Go up one row to R1 and set y = -2:
40. 40. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22From the bottom R2: -11y = 22  y = -2Go up one row to R1 and set y = -2:x + 4y = -7
41. 41. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22From the bottom R2: -11y = 22  y = -2Go up one row to R1 and set y = -2:x + 4y = -7x + 4(-2) = -7
42. 42. Matrix Notation2. Starting from the bottom row, get the answer for one of thevariable. Then go up one row, using the solution alreadyobtained to get another answer of another variable. Repeatthe process, working up the rows to extract all solutions.Example A. Solve using matrix notation. x + 4y = -7 Eq. 1{ 2x – 3y = 8 Eq. 2Put the system into a matrix: -2 -8 14 1 4 -7 -2*R1 add  R2 1 4 -7 2 -3 8 0 -11 22From the bottom R2: -11y = 22  y = -2Go up one row to R1 and set y = -2:x + 4y = -7x + 4(-2) = -7x=1 Solution : (1, -2)
43. 43. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3
44. 44. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix:
45. 45. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix:2 3 3 131 2 2 83 2 3 13
46. 46. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix:2 3 3 13 R1 R21 2 2 83 2 3 13
47. 47. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix:2 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13
48. 48. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix:2 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -2R1 add R2
49. 49. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -2R1 add R2
50. 50. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -2R1 add R21 2 2 80 -1 -1 -33 2 3 13
51. 51. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -2R1 add R21 2 2 80 -1 -1 -3 -3* R1 add R33 2 3 13
52. 52. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R21 2 2 80 -1 -1 -3 -3* R1 add R33 2 3 13
53. 53. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R21 2 2 8 1 2 2 80 -1 -1 -3 -3* R1 add R3 0 -1 -1 -33 2 3 13 0 -4 -3 -11
54. 54. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R21 2 2 8 1 2 2 80 -1 -1 -3 -3* R1 add R3 0 -1 -1 -33 2 3 13 0 -4 -3 -11 -4*R2 add R3
55. 55. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R21 2 2 8 1 0 24 2 4 12 80 -1 -1 -3 -3* R1 add R3 0 -1 -1 -33 2 3 13 0 -4 -3 -11 -4*R2 add R3
56. 56. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R21 2 2 8 1 0 24 2 4 12 80 -1 -1 -3 -3* R1 add R3 0 -1 -1 -33 2 3 13 0 -4 -3 -111 2 2 8 -4*R2 add R30 -1 -1 -30 0 1 1
57. 57. Matrix NotationExample B. Solve using matrix notation. E1{ 2x + 3y + 3z = 13 x + 2y + 2z = 8 E2 3x + 2y + 3z = 13 E3Put the system into a matrix: -2 -4 -4 -162 3 3 13 R1 R2 1 2 2 81 2 2 8 2 3 3 133 2 3 13 3 2 3 13 -3 -6 -6 -24 -2R1 add R21 2 2 8 1 0 2 4 2 12 4 80 -1 -1 -3 -3* R1 add R3 0 -1 -1 -33 2 3 13 0 -4 -3 -111 2 2 8 -4*R2 add R30 -1 -1 -30 0 1 1 Its in upper diagonal form. Ready to solve.
58. 58. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1
59. 59. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1From R3, we get z = 1.
60. 60. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1From R3, we get z = 1.From R2, -y – z = -3
61. 61. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1From R3, we get z = 1.From R2, -y – z = -3 -y – (1) = -3
62. 62. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1From R3, we get z = 1.From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=y
63. 63. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1From R3, we get z = 1,From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=yFrom R1, we getx + 2y + 2z = 8
64. 64. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1From R3, we get z = 1,From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=yFrom R1, we getx + 2y + 2z = 8x + 2(2) + 2(1) = 8
65. 65. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1From R3, we get z = 1,From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=yFrom R1, we getx + 2y + 2z = 8x + 2(2) + 2(1) = 8 x+6=8
66. 66. Matrix NotationWe have reduced the matrix.2 3 3 13 1 2 2 81 2 2 8 0 -1 -1 -33 2 3 13 0 0 1 1From R3, we get z = 1,From R2, -y – z = -3 -y – (1) = -3 3–1=y 2=yFrom R1, we getx + 2y + 2z = 8x + 2(2) + 2(1) = 8 x+6=8 x=2So the solution is (2, 2, 1)