M a t r i k s

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M a t r i k s

  1. 2. <ul><li>Understanding, N o tation, and Dimension of Matrix </li></ul><ul><li>Types of Matrix </li></ul><ul><li>Transpose and Similarity of a matrix </li></ul><ul><li>Operation of the Matrix </li></ul><ul><li>Inverse and Determinant of the Matrix </li></ul><ul><li>Completed the system of Linear Using Matrix </li></ul>
  2. 3. <ul><li>Understanding, Notation, and Demension Of Matrix </li></ul><ul><li>Understanding Of Matrix </li></ul><ul><li>Pay attention the following illustration </li></ul><ul><li>Mr. Andi note is student absent in last three year, that is January, February and March to 3 student that is Arlan, Bronto, and cery like at the table . </li></ul>On the table can be written : Arlan Bronto Cery January February March 3 4 1 6 2 3 1 2 4
  3. 4. is a rectangular array of numbers, consists of rows and columns and is written using brackets or parentheses. The entries of a matrix are called elements of matrix . An element of a matrix is addressed by listing the row number and then column number M A T R I X
  4. 5. Matrix is generally notated using capital latter
  5. 6. 2. The order of the matrix A matrix of A has m rows and n column is called as matrix of dimension on order m x n, and so notated of “A(mxn)”. To more understand the definition of the element of a matrix.
  6. 7. The first column The second column The third column The column n-th The second row The first row The third row The row n-th
  7. 8. Example: Matrix A = The first row The second row The first column The second column The third column <ul><ul><li>The order matrix A is 2 x 3 </li></ul></ul><ul><li>4 is the second row and the first column </li></ul>
  8. 9. a row matrix Is a matrix that only has a row A = ( 1 3 5), and B = ( -1 0 4 7) The order matrix is and
  9. 10. a column matrix Is a matrix that only has a column
  10. 11. A matrix square A square matrix a matrix has the number of row of a matrix equals the number of its column
  11. 12. Example : rows 4, columns 4 A is matrix the order 4 A = Main diagonal
  12. 13. A = Upper Triangle Matrix is square matrix which all of the element under the diagonal is zero Upper Triangle Matrix
  13. 14. B = B is a lower triangle matrix is square matrix which all of the element upper the diagonal is zero Lower Triangle Matrix
  14. 15. C = Diagonal Matrix is square matrix that all of element is zero, except the element on the diagonal not all of them Diagonal Matrix:
  15. 16. I = I is matrix Identity that is diagonal matrix that elements at main diagonal value one Pay attention the following matrix
  16. 17. <ul><li>Transpose and Similarity of a Matrix </li></ul><ul><li>Transpose of a Matrix </li></ul><ul><li>Let A is a matrix whit dimension of (m x n). From the matrix of A we can formed a new matrix that obtained by following method: </li></ul><ul><li>a. Change the line of i th of matrix A to the row of </li></ul><ul><li> ith of new matrix </li></ul><ul><li>b. Change the row of j th of matrix A to the line of </li></ul><ul><li> jth of new matrix </li></ul><ul><li>The new matrix that resulted is called transpose from matrix of A symbolized with A’ or From the above changess, the dimension of A’ is (n x m) </li></ul>
  17. 18. Transpose matrix A A = IS A t =
  18. 19. Example :
  19. 20. let A = (aij) ang B = (bij) are two matrices with the same dimension. Matrix of A is callled equal with matrix of B id the element that located on the two matrices has the same value. 2. Similarity of two matrix
  20. 21. One located element with the same value One located element with the same value One located element with the same value One located element with the same value
  21. 22. and B = A = If Matrix A = Matrix B, so x – 7 = 6  x = 13 2y = -1  y = - ½
  22. 23. Example 1: Given that K = And L = If K = L, find the value r?
  23. 24. Answer K = L = p = 6; q = 2p  q = 2.6 = 12 3r = 4q  3r = 4.12 = 48 jadi r = 48 : 3 = 16
  24. 25. Taking example A = and B = if A t = B, then determine the value x? Example 2:
  25. 26. Answer : A = = A t = B A t =
  26. 27. x + y = 1 x – y = 3 2x = 4 so x = 4 : 2 = 2 
  27. 28. <ul><li>Algebraic Operation on Matrix </li></ul><ul><li>Addition and Subtraction of Matrix </li></ul><ul><li>Scale Multiplication with a Matrix </li></ul><ul><li>Matrix Multiplication with Matrix </li></ul>
  28. 29. Addition/Subtraction  Two matrix can be summed/reduced if the order of the matrix are same and its statement in one position
  29. 30. Example 1: and B = A = A + B = + =
  30. 31. If A = , B = and C = hence(A + C) – (A + B) =…. Example 2:
  31. 32. (A + C) – (A + B) = A + C – A – B = C – B =  = = Answer
  32. 33. Scale Multiplication With a Matrix  Let k Є R and A is a matrix with dimension of m x n . Multiplication of real number k by matrix of A is a new matrix which is also has dimension of m x n that obtained by multiplying each element A by real number of k and notates kA
  33. 34. Matrix A = Determine matrix represented by 3A 3A = Example :1
  34. 35. Given Matrix of A = , B = and C = if A – 2B = 3C, So determine a + b ? Example 2 :
  35. 36. = 3 – = A – 2B = 3C – 2 Answer:
  36. 37. – = =
  37. 38. = a – 2 = -3  a = -1 4 – 2a – 2b = 6 4 + 2 – 2b = 6 6 – 2b = 6 -2b = 0  b = 0 Become a + b = -1 + 0 = -1
  38. 39. Matrix Multiplication with Matrix  The Product Of Two Matrices A and B can be got when satisfies the relation   A m x n = B p x q = AB m x q   Equal
  39. 40. The number of column of matrix A should equal the number of rows of matrix B, the product, that is AB has order of m x q. when m is the number of rows of matrix A and q is the number of column of matrix B
  40. 41. 26 November 2011 The second row The first row The second column row 1 x column 1 row 1 x column 2 row 2 x column 1 row 2 x column 2 The first column = x … … … …………… row 1 x……. ……… .x column1 A m x n x B n x p = C m x p …………… .. ………… ..
  41. 42. 26 November 2011 3 4 1 2 7 8 1 x 5 + 2 x 6 1 x 7 + 2 x 8 3 x 5 + 4 x 6 3 x 7 + 4 x 8 5 6 = x Example 1:
  42. 43. 26 November 2011 1 x 5 + 2 x 6 1 x 7 + 2 x 8 3 x 5 + 4 x 6 3 x 7 + 4 x 8 = = 17 23 39 53
  43. 44. 26 November 2011 6 8 5 7 2 4 5 x 1 + 7 x 3 5 x 2 + 7 x 4 6 x 1 + 8 x 3 6 x 2 + 8 x 4 1 3 = x = 26 38 30 44 Example 2:
  44. 45. 26 November 2011 A = Determine: A x B and B x A and B = Example 3 :
  45. 46. 26 November 2011 A x B = = = 3 x 5 + (-1) x 8 2 x (-2) + 4 x 1 2 x 5 + 4 x 8 3 x (-2) + (-1) x 1 3 2 4 -1 3 2 4 -1 3 2 4 -1 -2 5 1 8 -7 7 0 42
  46. 47. 26 November 2011 = 4 (-2) x (-1) + 5 x 4 1 x 3 + 8 x 2 1 x (-1) + 8 x 4 (-2) x 3 + 5 x 2 = 22 19 31 B x A = 3 2 4 -1 -2 5 1 8
  47. 48. 26 November 2011 conclusion A x B  B x A That is not satisfies the commutative charecteristics
  48. 49. Determinant of a Matrix Determinant of a Matrix with Dimension of 2 x 2 Determinant from a matrix of A notated with det (A), , or is a certain value with the size is equal (ad – bc)
  49. 50. Example 1: Determine the determinant of following matrix!
  50. 51. Answer :
  51. 52. Example 2:
  52. 53. Answer : = = = = = 3 (2x)(-3) – (x – 7)(3) 3 3 -6x – 3x + 21 -18 2 -9x x
  53. 54. Determinant of a Matrix with Dimension of 3 x 3 + + + - - -
  54. 55. Determine the determinant of following matrix! Example 2:
  55. 56. Answer : (-3.2.-7)+(0.13.4)+(5.-5.0)-(5.2.4)- (-3.13.0)+(0.-5.-7) = 42 + 0 + 0 – 40 – 0 – 0 42 - 40 2 = =
  56. 57. 26 November 2011 <ul><li>Inverse Matrix </li></ul><ul><li>If A and B are two square matrices with the same dimension such that satisfies AB = BA = I where I is an identity matrix, then </li></ul><ul><li>Matrix of A called inverse from matrix of B given notation of B </li></ul><ul><li>Matrix of B called inverse from matrix of A given notation of A </li></ul>-1 -1
  57. 58. 26 November 2011 A = and B = A x B = = -5+6 -3+3 10-10 6-5 = = <ul><ul><li>I </li></ul></ul>Example 1:
  58. 59. 26 November 2011 A = and B = B x A = = -5+6 -15+15 2-2 6-5 = = <ul><ul><li>I </li></ul></ul>Example 2
  59. 60. 26 November 2011 Inverse Matrix (2 x 2) if A = Then inverse matrix of A is A -1 =
  60. 61. 26 November 2011 If determinant of a matrix equals zero then the matrix do not have inverse. This kind of matrix is called singular matrix
  61. 62. 26 November 2011 Determine the inverse following of a Matrix! A = Example
  62. 63. 26 November 2011 3 2 -1 -5 Answer
  63. 64. 26 November 2011 Characteristics inverse of matrix: ( A. B ) -1 = B -1 . A -1 ( A -1 ) -1 = A A .A -1 = A -1 . A = I 1. 2. 3.
  64. 65. 26 November 2011 Example : Given that A = and B = so (AB) -1 is….
  65. 66. 26 November 2011 AB = Answer
  66. 67. 26 November 2011
  67. 68. Completed the system of linear Equation Using Matrix <ul><li>There are two methods can be used to determined the solution of system of linear equation using the matrix approach that are: </li></ul><ul><li>Determinant method </li></ul><ul><li>Inverse matrix method </li></ul>
  68. 69. Variables <ul><li>Using determinant method, the value of x and y found out by the following formula </li></ul>
  69. 70. Variables b. Using inverse of matrix method, the value of x and y found out by using the following steps.
  70. 71. Thank You

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