Section 4.3 Day 1 Determinants
Determinant of order 2 Consider a 22 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:
Finding Determinants of Matrices Notice the different symbol: the straight lines tell you to find the determinant!! - (-5 * 2) = (3 * 4) 12 - (-10) = 22 =
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
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
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.
Formula • Once the matrix has been written, use the formula: • Where indicates that the appropriate sign should be chosen to yield a positive value.
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
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
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
Assignment Section 4.3 (Day 1): page 218 – 219 # 12 – 33 (÷3)
Section 4.3 Day 2 Cramer’s Rule
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:
Let A = • If det A ≠0, then the system has exactly one solution, which is found by:
Use Cramer’s rule to solve the system: 1) 8x + 5y = 2 2x – 4y = -10 Det A = = 8(-4) – (5)(2) = -32 – 10 = -42
Example continued X = Y = X = Y = X = -1 Y = 2 (-1,2)
Cramer’s Rule for 3 equations (3 variables) • Linear equation: ax + by + cz = j dx + ey + fz =k gx + hy + iz = L Coefficient matrix:
Let A = • If det A ≠0, then the system has exactly one solution, which is found by:
Use Cramer’s rule to solve the system: 1) x + 3y – z = 1 -2x – 6y + z = -3 3x + 5y – 2z = 4 Det A = = -4
Example continued = 2
Example continued = 0
Example continued = 1 (x,y,z) = (2, 0, 1)
Assignment Section 4.3 (Day 2) Page 218 – 219 # 36 – 51 (÷ 3), 55 – 56, 62 - 63
