1 © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved Chapter 9 Matrices and Determinants

OBJECTIVES © 2010 Pearson Education, Inc. All rights reserved 2 The Matrix Inverse Verify the multiplicative inverse of a matrix. Find the inverse of a matrix. Find the inverse of a 2 × 2 matrix. Use matrix inverses to solve systems of linear equations. Use matrix inverses in applied problems. SECTION

3 © 2010 Pearson Education, Inc. All rights reserved INVERSE OF A MATRIX If A be an n × n matrix and let I be the n × n identity matrix that has 1’s on the main diagonal and 0s elsewhere. If there is an n × n matrix B such that then B is called the inverse of A and we write B = A –1 (read “A inverse”).

4 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Verifying the Inverse of a Matrix Show that B is the inverse of A: Solution You need to verify that AB = I and BA = I.

5 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Verifying the Inverse of a Matrix Solution continued

6 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 1 Verifying the Inverse of a Matrix Solution continued Since AB = I and BA = I it follows that B = A −1.

7 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Proving That a Particular Nonzero Matrix Has No Inverse Show that the matrix A does not have an inverse. Solution Suppose A has an inverse B, where

8 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 2 Proving That a Particular Nonzero Matrix Has No Inverse Solution continued Then you have AB = I Multiply the two matrices on the left side. Since these two matrices are equal, you must have 0 = 1. Because this is a false statement, the matrix A does not have an inverse.

9 © 2010 Pearson Education, Inc. All rights reserved PROCEDURE FOR FINDING THE INVERSE OF A MATRIX Let A be an n × n matrix. 1.Form the n × 2n augmented matrix [A|I], where I is the n × n identity matrix. 2.If there is a sequence of row operations that transforms A into I, then this same sequence of row operations will transform [A|I] into [I|B], where B = A –1. 3.Check your work by showing that AA –1 = I.

10 © 2010 Pearson Education, Inc. All rights reserved PROCEDURE FOR FINDING THE INVERSE OF A MATRIX If it is not possible to transform A into I by row operations, then A does not have an inverse. (This occurs if, at any step in the process, you obtain a matrix [C|D] in which C has a row of zeros.)

11 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Inverse of a Matrix Find the inverse (if it exists) of the matrix Solution Step 1Start with the matrix

12 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Inverse of a Matrix Solution continued Step 2Use row operations.

13 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Inverse of a Matrix Solution continued

14 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 4 Finding the Inverse of a Matrix Solution continued

15 © 2010 Pearson Education, Inc. All rights reserved A RULE FOR FINDING THE INVERSE OF 2 × 2 MATRIX If ad – bc = 0, the matrix does not have an inverse. The matrix if and only if ad – bc ≠ 0. Moreover, if ad − bc ≠ 0, then the inverse is given by is invertible

16 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Inverse of a 2 × 2 Matrix Find the inverse (if it exists) of each matrix. Solution a. For the matrix A, a = 5, b = 2, c = 4, and d = 3. Here ad – bc = (5)(3) – (2)(4) = 15 – 8 = 7 ≠ 0 so A is invertible.

17 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 5 Finding the Inverse of a 2 × 2 Matrix Solution continued A −1 = You should verify that AA −1 = I. b. For the matrix B, a = 4, b = 6, c = 2, and d = 3. Here ad – bc = (4)(3) – (6)(2) = 0 so B does not have an inverse.

18 © 2010 Pearson Education, Inc. All rights reserved SOLVING SYSTEMS OF LINEAR EQUATIONS BY USING MATRIX INVERSES Matrix multiplication can be used to write a system of linear equations in matrix form. The solution to this equation is Solving a system of linear equations amounts to solving the matrix equation of the form

19 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solving a Linear System by Using an Inverse Matrix Use a matrix to solve the linear system. Solution Write the linear system in matrix form. Use zeros for coefficients of missing variables. A X = B

20 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solution continued Since the matrix is invertible (see Example 4) the system has a unique solution X = A –1 B. Solving a Linear System by Using an Inverse Matrix Computed in Example 4

21 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 6 Solution continued The solution set is {(3, 1, 4)}, which you can check in the original system. Solving a Linear System by Using an Inverse Matrix

22 © 2010 Pearson Education, Inc. All rights reserved THE LEONTIEF INPUT-OUTPUT MODEL Suppose a simplified economy depends on two products: energy (E) and food (F). To produce one unit of E requires unit of E and unit of F. To produce one unit of F requires unit of E and unit of F. Then the interindustry consumption is given by the matrix to the right.

23 © 2010 Pearson Education, Inc. All rights reserved THE LEONTIEF INPUT-OUTPUT MODEL The matrix A is the interindustry technology input-output matrix, or simply the technology matrix, of the system. If the system is producing x 1 units of energy and x 2 units of food, then the column matrixis called the gross production matrix.

24 © 2010 Pearson Education, Inc. All rights reserved THE LEONTIEF INPUT-OUTPUT MODEL Consequently, represents the interindustry consumption. units consumed by E units consumed by F

25 © 2010 Pearson Education, Inc. All rights reserved THE LEONTIEF INPUT-OUTPUT MODEL where I is the identity matrix. The gross production matrix is: X = (I – A) –1 D represents consumer demand, then If the column matrix

26 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using the Leontief Input-Output Model In the preceding discussion, suppose the consumer demand for energy is 1000 units and that for food is 3000 units. Find the level of production (X) that will meet interindustry and consumer demand. Solution For the matrix A, we have

27 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using the Leontief Input-Output Model Use the formula for the inverse of a 2 × 2 matrix Solution continued

28 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using the Leontief Input-Output Model Hence, the gross production matrix X is given by Solution continued

29 © 2010 Pearson Education, Inc. All rights reserved EXAMPLE 7 Using the Leontief Input-Output Model Thus, to meet the consumer demand for 1000 units of energy and 3000 units of food, the Solution continued energy produced must be food production must be units and units.