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3.8 Use Inverse Matrices to Solve Linear Systems

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Presentation on theme: "3.8 Use Inverse Matrices to Solve Linear Systems"— Presentation transcript:

1 3.8 Use Inverse Matrices to Solve Linear Systems

2 The number 1 Is the multiplicative identity for real numbers because 1*a=a and a*1=a for all real numbers a. For matrices, the n x n identity matrix is the matrix that has 1’s on the main diagonal and 0’s elsewhere. 2 x 2 Identity Matrix 3 x 3 Identity Matrix If A is any n x n matrix and I is the n x n identity matrix, then IA=A and AI=A

3 Two n x n matrices are inverses of each other if their product (in both orders) is the n x n identity matrix. For example, matrices A and B about to appear are inverses of each other. We will use this symbol to denote the inverse of matrix A

4 The Inverse of a 2 x 2 Matrix
The inverse of matrix Provided ad – cb is not equal to zero.

5 A I = I A = A Verify the identity property above indicated for matrix A below: 8 2 6 4 A= 1 I= Diagonal 1 8 2 6 4 I A = = 1(8) + 0(6) 1(2) + 0(4) 0(8) + 1(6) 0(2) + 1(4) 8 2 6 4 = 1 8 2 6 4 A I = = 8(1) + 6(0) 2(1) + 4(0) 8(0) + 6(1) 2(0) + 4(1) 8 2 6 4 =

6 Calculate the A for matrix A:
-1 8 2 6 4 A= 8 2 6 4 = (8)(4) –(6)(2) =32 -12 =20 = 4 20 -2 -6 8 = 1 5 -1 10 -3 2 4 -2 -6 8 1 20 A = -1

7 Calculate the A for matrix A:
-1 4 3 5 1 A= 4 3 5 1 = (4)(1) –(5)(3) =4 -15 = -11 = -1 11 3 5 -4 1 -3 -5 4 1 -11 A = -1

8 Write the system of equations represented by each matrix equation:
-3 6 7 1 x y = 15 -8 Notice that this matrix equation could be solved for X by multiplying both sided of the equation by the Inverse Matrix A -3x + 6y = 15 7x + y = -8 5 9 -2 4 x y = 5x + 9y = 0 -2x + 4y = 5

9 Multiplying both sides by the inverse:
Solve the following system of equations using a Matrix Equation: 4x + 2y = 10 5x + y = 17 1 Multiplying both sides by the inverse: Write as matrix equation: -1 6 1 3 5 -2 = -1 6 1 3 5 -2 4 2 5 1 x y = 10 17 4 2 5 1 x y 10 17 Finding the determinant of the coefficient matrix: 4 2 5 1 = (4)(1) –(5)(2) =4 -10 = -6 Finding the inverse of the coefficient matrix: = -1 6 2 5 -4 = -1 6 1 3 5 -2 1 -2 -5 4 1 -6

10 4x + 2y = 10 5x + y = 17 1 Write as matrix equation: -1 6 1 3 5 -2 -1 6 1 3 5 -2 4 2 5 1 x y 10 17 4 2 5 1 x y 10 17 = = -1 6 1 3 (4) (5) + -1 6 1 3 (2) (1) + -1 6 1 3 (10) (17) + x y = 5 6 -2 3 (10) (17) + 5 6 -2 3 (4) (5) + 5 6 -2 3 (2) (1) + 1 x y = 4 -3 x y = 4 -3 Solution is (4,-3)

11 Multiplying both sides by the inverse:
Solve the following system of equations using matrices: 2x + 5y = 13 6x + 3y = 3 Multiplying both sides by the inverse: Write as matrix equation: -1 8 5 24 1 4 12 = -1 8 5 24 1 4 12 2 5 6 3 x y = 13 3 2 5 6 3 x y 13 3 Finding the determinant of the coefficient matrix: 2 5 6 3 = (2)(3) –(6)(5) =6 -30 = -24 Finding the inverse of the coefficient matrix: = -3 24 5 6 -2 = -1 8 5 24 1 4 12 3 -5 -6 2 1 -24

12 Solve the following system of equations using matrices:
2x + 5y = 13 6x + 3y = 3 Write as matrix equation: -1 8 5 24 1 4 12 -1 8 5 24 1 4 12 2 5 6 3 x y 13 3 2 5 6 3 x y 13 3 = = -1 8 5 24 (2) (6) + -1 8 5 24 (5) (3) + -1 8 5 24 (13) ( 3) + = x y 1 4 -1 12 (13) ( 3) + 1 4 -1 12 (2) (6) + 1 4 -1 12 (5) (3) + 1 x y = -1 3 x y = -1 3 Solution is (-1,3)


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