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Section 4.3

Section 4.3. Day 1 Determinants. Determinant of order 2. Consider a 2  2 matrix:. Determinant of A , denoted , is a number and can be evaluated by. Determinant of order 2. easy to remember (for order 2 only). +. -. Example: Evaluate the determinant:.

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Section 4.3

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  1. Section 4.3 Day 1 Determinants

  2. Determinant of order 2 Consider a 22 matrix: • Determinant of A, denoted , is a number and can be evaluated by

  3. Determinant of order 2 • easy to remember (for order 2 only). + - Example: Evaluate the determinant:

  4. Finding Determinants of Matrices Notice the different symbol: the straight lines tell you to find the determinant!! - (-5 * 2) = (3 * 4) 12 - (-10) = 22 =

  5. Determinants of order 3 Consider an example: Its determinant can be obtained by: You are encouraged to find the determinant by using other rows or columns

  6. Finding Determinants of Matrices 2 0 1 -2 -1 4 = [(2)(-2)(2) + (0)(5)(-1) + (3)(1)(4)] [(3)(-2)(-1) + (2)(5)(4) + (0)(1)(2)] - = - [-8 + 0 +12] [6 + 40 + 0] = 4 – 6 - 40 = -42

  7. Area of a Triangle, Using Determinants • It turns out that the area of a triangle can also be found using determinants. • What you do is form a 3×3 determinant where the first column are the x's for all the points, the second column are the y's for all the points, and the last column is all ones.

  8. Formula • Once the matrix has been written, use the formula: • Where indicates that the appropriate sign should be chosen to yield a positive value.

  9. Find the area of the triangle with coordinates: (1,2), (6,2) and (4,0) Determinant = [(1x2x1) + (2x1x4) + (1x6x0)] – [(1x2x4) + (1x1x0) + (2x6x1)] = [2+8+0] – [8+0+12] = 10 – 20 = -10 Area = ± ½ (-10) = 5

  10. Estimate the area of the Bermuda Triangle if the approximate coordinates are: (938,454), (900, -518) and (0,0) D= [(938x-518x1)+(454x1x0) + (1x900x0)] – [(1x-518x0)+(938x1x0) + (454x900x1)] = [-485,884+0+0] – [0+0+408,600] = -485,884 - 408,600 = -894,484 Area = ± ½ (-894,484) = 447,242

  11. Finding the Determinant Using the Calculator • Entering the matrix: 2nd, MATRIX, , • Getting the determinants 􀂾 2nd, QUIT (back to home screen) 􀂾 2nd, MATRIX, 􀂾 Down to “det”, ENTER 􀂾 2nd, MATRIX, ENTER (for A), “)”, ENTER

  12. Assignment Section 4.3 (Day 1): page 218 – 219 # 12 – 33 (÷3)

  13. Section 4.3 Day 2 Cramer’s Rule

  14. Cramer’s Rule for 2 equations (2 variables) • Uses determinants to solve a system of linear equations • Uses the coefficient matrix • Linear equation: ax + by = e cx + dy = f Coefficient matrix:

  15. Let A = • If det A ≠0, then the system has exactly one solution, which is found by:

  16. Use Cramer’s rule to solve the system: 1) 8x + 5y = 2 2x – 4y = -10 Det A = = 8(-4) – (5)(2) = -32 – 10 = -42

  17. Example continued X = Y = X = Y = X = -1 Y = 2 (-1,2)

  18. Cramer’s Rule for 3 equations (3 variables) • Linear equation: ax + by + cz = j dx + ey + fz =k gx + hy + iz = L Coefficient matrix:

  19. Let A = • If det A ≠0, then the system has exactly one solution, which is found by:

  20. Use Cramer’s rule to solve the system: 1) x + 3y – z = 1 -2x – 6y + z = -3 3x + 5y – 2z = 4 Det A = = -4

  21. Example continued = 2

  22. Example continued = 0

  23. Example continued = 1 (x,y,z) = (2, 0, 1)

  24. Assignment Section 4.3 (Day 2) Page 218 – 219 # 36 – 51 (÷ 3), 55 – 56, 62 - 63

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