advertisement

Aim: Determinants and Cramer’s law Do Now: 1. Solve for x and y in terms of a,b,c,d,e,f Use elimination to eliminate y and solve for x and vice versa ax + by = c dx + ey = f (ce - bf) x= (ae - bd) 2. Solve for x and y in terms of a,b,c,d,e,f,g,h,i,j,k,l ax + by +cz = d ex + fy + gz = h ix + jy + kz = l Determinant of: =a bG = ad - bc c d =3 7 G = (3 # 10) - (7 # 5) = - 5 5 10 To find the determinant of a matrix with a calculator: Matrix type in matrix quit matrix math det insert the matrix enter Cramer’s Rule: 3x – 2y = 4 6x + y = 13 =3 - 2 G = (3 # 1) - (- 2 # 6) = 15 D= 6 1 Dx = replace column x with the answers to the equations: =4 - 2 G = (4 # 13) - (- 2 # 13) = 30 Dx = 13 1 Dy = replace column y with the answers to the equations: =3 4 G = (3 # 13) - (4 # 6) = 15 Dy = 6 13 Cramer’s law states that the solutions of Dx D Dy = y D x= So in this case: 30 = 2 15 15 = y= 1 15 x= Cramer’s Rule for 3 by 3 Systems of Equations: 3x + y – z = 2 -x +2y + 4z = 15 x – 2y -3z = -12 To solve a 3 by 3 determinant you do something called expand along the first row. Take the first number and then times it by the determinant of the 2 by 2 matrix left after the row and column the first number is in is crossed out. Also remember to alternate signs. >3 D = - 1 2 > =+ 3# 2 1 - 1 15 2 4 - 12 - 2 - 3 3 2 - 1 - 1 15 4 1 - 12 - 3 > Dy = 4 H =2 - 2 - 3 1 Dx = - 1 1 4 - 2 - 3 H = 7 H = 14 G =- 1 4 G =- 1 - +- 1 1 3 1 2 - 2 G = 7 > Dz = 3 1 2 - 1 2 15 1 - 2 - 12 H = 21 After applying Cramer’s rule we get Dx = D Dy = = y D Dz = z= D x= Doris Xia 7 = 1 7 14 = 2 7 21 = 3 7