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MATRICES

MATRICES. After watching the slides show, you will be able to Determine the solution of matrix problem by using Multiple matrix operation and inverse of matrix and its characteristics. Multiply matrix with matrix Look at this illustration: Randy and Lya want to buy

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MATRICES

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  1. MATRICES

  2. After watching the slides show, you will be able to Determine the solution of matrix problem by using Multiple matrix operation and inverse of matrix and its characteristics.

  3. Multiply matrix with matrix Look at this illustration: Randy and Lya want to buy book and pencil. Randy buys 3 books and 1 pencil. Lya buys 4 books and 2 pencils.

  4. If the price of a book is Rp500,00 and a pencil is Rp150,00; How much money each of them has to pay?

  5. Solution: Randy = 3 x 500 + 1 x 150 = Rp1.650,00 Lya = 4 x 500 + 2 x 150 = Rp2.300,00 The problem can be solved by using this matrix multiply:

  6. 3 1 500 4 2 150 (2 x 2) (2 x 1) column= row 3 x 500 + 1 x 150 = 4 x 500 + 2 x 150 1650 = 2300 (2 x 1)

  7. Condition of Matrix Multiply Matrix A can be multiplied with matrix B if Number of matrix A column = Number of matrix B row

  8. If matrix A has ordo m x n and matrix B has ordo n x p then A x B = C with C has ordo m x p Am x n x Bn x p = Cm x p

  9. How to Multiply Matrix if A x B = C then matrix element of C is the sum of multiple result of Element of row matrix A with element of column matrix B which is accordance

  10. Am x n x Bn x p = Cm x p 2nd C O L U M N …………… 1st C O L U M N 1st row x Baris 2 … … … 1st row x 1st column 1st row x2nd column 1st row x…. = 2nd row x 1st column 2nd row x2nd column ………….. …………….. ……….x 1st column

  11. Example 1: 1 2 5 6 7 8 x 3 4 1 x 5 + 2 x 6 1 x 7 + 2 x 8 = 3 x 5 + 4 x 6 3 x 7 + 4 x 8

  12. 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

  13. Example 2: 5 7 1 3 2 4 x 6 8 5 x 1 + 7 x 3 5 x 2 + 7 x 4 = 6 x 1 + 8 x 3 6 x 2 + 8 x 4 26 38 = 30 44

  14. Example 3: Given that: A = and B = Evaluate: A x B and B x A

  15. -1 -1 -1 3 3 3 -2 5 2 2 2 4 4 4 1 8 -7 7 0 42 A x B = 3 x (-2) + (-1) x 1 3 x 5 + (-1) x 8 = 2 x (-2) + 4 x 1 2 x 5 + 4 x 8 =

  16. -2 5 -1 3 B x A = 1 8 2 4 (-2) x 3 + 5 x 2 (-2) x (-1) + 5 x 4 = 1 x 3 + 8 x 2 1 x (-1) + 8 x 4 4 22 = 19 31

  17. Conclusion: A x B  B x A It means matrix multiply is not commutative

  18. Example 4: The value of a from the matrices equation: is…. + =

  19. Discuss: -1 -1 d 4 -5 2 2c 1 + = a +1 c 3 -4 -b 3 b -3 4c + (-c) 2 + (-1)(a + 1) 3 d - 5 = -b - 3 -8c + 3c -4+ 3(a + 1) 3 + b =

  20. 3 = 3c  c = 1 -b – 3 = -5c -b – 3 = -5 -b = -2  b = 2 3 + b = -1 + 3a 3 + 2 = -1 + 3a 5 = -1 + 3a 6 = 3a So, the value of a is 2

  21. Matrices Inverse Definition: If result of multiply two matrices is identity matrix, (A x B = B x A = I) then Matrix of A is matrix inverse of B or reverse matrix of B is matrix inverse of A

  22. Example 1 A = and B = A x B = -5+6 -3+3 = 10-10 6-5 = = I

  23. Example 2 A = and B = B x A = -5+6 -15+15 = 2-2 6-5 = I =

  24. because A x B = B x A = I it means B = inverse of A, or A = inverse of B. If B = inverse of A and written as A-1 then A. A-1=A-1. A = I

  25. Inverse Matrices (2 x 2) If A = then inverse of matrix A is A-1 = ad – bc = determinant matrix A d -b -c a

  26. If ad – bc = 0 it means That matrix does not have any inverse. The matrix that does not have any inverse is called singular matrix

  27. Example: Given that A = Find the inverse of matrix A

  28. Discuss: 3 -1 -5 2

  29. Inverse Matrices Characteristics: 1. A.A-1 = A-1.A= I 2. (A. B)-1 = B-1. A-1 3. (A-1 )-1 = A

  30. Example 1 Given that A = and B = then (AB)-1 is … .

  31. Discuss: AB = -2 + 6 0 - 2 0 - 4 -6 + 12

  32. -4 2 -6 4

  33. Example 2 If inverse of matrix A = then find matrix A

  34. Discuss: A = (A-1 )-1 2 -1 3 -4

  35. The solution of Matrices Equation If A, B and M are matrices which has ordo(2x2) and A is non singular matrix then the solutions of matrices equation are ☻AM = B is M = A-1.B ☺MA = B is M = B.A-1

  36. Example 1 Given that A = and B = Find M if: a. AM = B b. MA = B

  37. Discuss:

  38. If AM = B • then M = A-1.B

  39. b. If MA = B then M = B.A-1

  40. Example 2 Given that multiple matrices The value of a + b + c + d is ….

  41. Discuss:

  42. find a = 1, b = -3, c = 4 and d = 5 means a + b + c + d = 1 – 3 + 4 + 5 = 7

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