Cramers Rule (for solution of 3x3 linear equations)

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Cramers Rule (for solution of 3x3 linear equations)

  1. 1. Matrix Lesson 2: Cramer’s rule Dave Coulson, 2013
  2. 2. Three men take turns standing in pairs on a weighing machine. Alan and Barry together weigh 140kg. Alan and Charlie together weigh 110kg. Barry and Charlie together weigh 90kg. How heavy are the three men individually?
  3. 3. In maths, the problem looks like this: A + B = 140 A + C = 110 B + C = 90 This is a 3x3 system of linear equations. “3x3” because there are three equations and three unknowns. “Linear” because the unknowns are simply being added together. There are no squares or square roots or other peculiar functions.
  4. 4. I‟m going to solve this system using a procedure called Cramer’s Rule. A + B = 140 A + C = 110 B + C = 90 It‟s quite longwinded, but I like it because it produces the answer in a seemingly magical way. For most of the journey you won‟t know what you‟re doing or why you‟re doing it. but then the answer suddenly pops out in the most unlikely way.
  5. 5. First, rewrite the system in matrix notation: = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 A + B = 140 A + C = 110 B + C = 90
  6. 6. Then get the determinant of the matrix. = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 Detm = -2 1 1 0 1 1 1 0 1 1 0 0 1 1 0 1 (0 + 0 + 0) - (0 + 1 + 1)
  7. 7. Do it again, but this time replace the „A‟ row of the matrix with the numbers from the right-hand-side of the equation. = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 Detm = -2 140 1 0 140 1 110 0 1 110 0 90 1 1 90 1 DetA = -160
  8. 8. Detm = -2 A = 80kg = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 DetA = -160 The value for A is the fraction made from these two numbers (with DetA on top).
  9. 9. Detm = -2 B and C can be worked out in the same sort of way. Replace the „B‟ line of the matrix with the numbers on the right hand side of the equation. = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 (next screen)
  10. 10. Detm = -2 = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 1 140 0 1 140 1 110 1 1 110 1 90 1 0 90 DetB = -120
  11. 11. Detm = -2 = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 DetB = -120 B = 60kg
  12. 12. 1 1 140 1 1 1 0 110 1 0 0 1 90 0 1 = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 Detm = -2 DetC = -60
  13. 13. = 1 1 0 1 0 1 0 1 1 A B C 140 110 90 Detm = -2 DetC = -60 C = 30kg
  14. 14. Always check the answers against the initial clues: Alan and Barry together weigh 140kg. Alan and Charlie together weigh 110kg. Barry and Charlie together weigh 90kg. 80 + 60 80 + 30 60 + 30 (Charlie is obviously a child)
  15. 15. [END] Dave Coulson, 2013 According to Wikipedia, the technique was established by Gabriel Cramer (a Swiss mathematician) in 1750, “although Colin MacLaurin [Scottish] also published special cases of the rule in 1748 (and possibly knew of it as early as 1729).”

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