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Chapter 9 Matrices and Determinants © 2010 Pearson Education, Inc. All rights reserved © 2010 Pearson Education, Inc. All rights reserved 1 SECTION 9.3 The Matrix Inverse OBJECTIVES 1 2 3 4 5 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. © 2010 Pearson Education, Inc. All rights reserved 2 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 AB I and BA I , then B is called the inverse of A and we write B = A–1 (read “A inverse”). © 2010 Pearson Education, Inc. All rights reserved 3 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. © 2010 Pearson Education, Inc. All rights reserved 4 EXAMPLE 1 Verifying the Inverse of a Matrix Solution continued © 2010 Pearson Education, Inc. All rights reserved 5 EXAMPLE 1 Verifying the Inverse of a Matrix Solution continued Since AB = I and BA = I it follows that B = A−1. © 2010 Pearson Education, Inc. All rights reserved 6 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 © 2010 Pearson Education, Inc. All rights reserved 7 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. © 2010 Pearson Education, Inc. All rights reserved 8 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. © 2010 Pearson Education, Inc. All rights reserved 9 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.) © 2010 Pearson Education, Inc. All rights reserved 10 EXAMPLE 4 Finding the Inverse of a Matrix Find the inverse (if it exists) of the matrix 1 1 0 A 0 3 1 . 2 3 3 Solution Step 1 Start with the matrix 1 1 0 1 0 0 A I 0 3 1 0 1 0 2 3 3 0 0 1 © 2010 Pearson Education, Inc. All rights reserved 11 EXAMPLE 4 Finding the Inverse of a Matrix Solution continued Step 2 1 1 1 R2 R2 0 1 3 2 R1 R3 R3 0 1 Use row operations. 0 1 1 0 3 3 2 0 0 1 0 3 0 1 1 1 0 1 0 0 1 1 0 0 1 R2 R3 R3 0 1 3 3 8 1 0 0 2 1 3 3 © 2010 Pearson Education, Inc. All rights reserved 12 EXAMPLE 4 Finding the Inverse of a Matrix Solution continued 1 0 1 1 0 1 0 1 1 0 3 3 3 6 1 R3 0 0 1 8 8 8 0 0 3 8 1 1 0 1 0 0 1 2 3 1 R3 R2 R2 0 1 0 3 8 8 8 6 1 3 0 0 1 8 8 8 © 2010 Pearson Education, Inc. All rights reserved 13 EXAMPLE 4 Finding the Inverse of a Matrix Solution continued 1 R2 R1 R1 1 0 0 1 0 0 3 1 6 8 8 8 6 3 2 3 1 1 1 A 2 3 8 8 8 8 6 1 6 1 3 8 8 8 © 2010 Pearson Education, Inc. All rights reserved 6 0 8 2 0 8 6 1 8 1 1 3 3 1 8 8 3 1 8 8 1 3 8 8 14 A RULE FOR FINDING THE INVERSE OF 2 × 2 MATRIX a b The matrix A is invertible c d if and only if ad – bc ≠ 0. Moreover, if ad − bc ≠ 0, then the inverse is given by 1 d b A . ad bc c a 1 If ad – bc = 0, the matrix does not have an inverse. © 2010 Pearson Education, Inc. All rights reserved 15 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. © 2010 Pearson Education, Inc. All rights reserved 16 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. © 2010 Pearson Education, Inc. All rights reserved 17 SOLVING SYSTEMS OF LINEAR EQUATIONS BY USING MATRIX INVERSES Matrix multiplication can be used to write a system of linear equations in matrix form. 3x 2y 4 3 2 x 4 4 3 y 5 4x 3y 5 Solving a system of linear equations amounts to solving the matrix equation of the form AX B. The solution to this equation is 1 X A B. © 2010 Pearson Education, Inc. All rights reserved 18 EXAMPLE 6 Solving a Linear System by Using an Inverse Matrix Use a matrix to solve the linear system. 4 x y 3y z 7 2 x 3 y 3z 21 Solution Write the linear system in matrix form. 1 1 0 x 4 Use zeros for 0 3 1 y 7 coefficients of missing variables. 2 3 3 z 21 A X = B © 2010 Pearson Education, Inc. All rights reserved 19 EXAMPLE 6 Solving a Linear System by Using an Inverse Matrix Solution continued Since the matrix is invertible (see Example 4) the system has a unique solution X = A–1B. Computed in Example 4 © 2010 Pearson Education, Inc. All rights reserved 20 EXAMPLE 6 Solving a Linear System by Using an Inverse Matrix Solution continued 6 3 1 4 1 1 X A B 2 3 1 7 8 6 1 3 21 6 4 3 7 1 21 24 3 1 1 2 4 3 7 1 21 8 1 8 8 6 4 1 7 3 21 32 4 The solution set is {(3, 1, 4)}, which you can check in the original system. © 2010 Pearson Education, Inc. All rights reserved 21 THE LEONTIEF INPUT-OUTPUT MODEL Suppose a simplified economy depends on two products: energy (E) and food (F). To produce 1 1 one unit of E requires unit of E and unit of F. 4 2 To produce one unit of F 1 requires unit of E and 3 1 unit of F. Then the 4 interindustry consumption is given by the matrix to the right. © 2010 Pearson Education, Inc. All rights reserved 22 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 x1 units of energy and x2 units of food, then the x1 column matrix X is called the gross x2 production matrix. © 2010 Pearson Education, Inc. All rights reserved 23 THE LEONTIEF INPUT-OUTPUT MODEL Consequently, 1 1 1 1 x x 1 2 4 3 x1 4 3 AX 1 1 x 1 1 2 x x 2 4 2 1 4 2 units consumed by E units consumed by F represents the interindustry consumption. © 2010 Pearson Education, Inc. All rights reserved 24 THE LEONTIEF INPUT-OUTPUT MODEL d1 If the column matrix D d2 represents consumer demand, then D X AX IX AX I A X I A 1 D I A 1 I A X IX X where I is the identity matrix. The gross production matrix is: X = (I – A)–1D © 2010 Pearson Education, Inc. All rights reserved 25 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 1 1 0 4 I A 0 1 1 2 1 3 3 4 1 1 4 2 © 2010 Pearson Education, Inc. All rights reserved 1 3 3 4 26 EXAMPLE 7 Using the Leontief Input-Output Model Solution continued Use the formula for the inverse of a 2 × 2 matrix 3 1 4 3 1 1 I A 3 3 1 1 1 3 4 4 3 2 2 4 3 48 4 19 1 2 1 3 1 36 16 3 19 24 36 4 © 2010 Pearson Education, Inc. All rights reserved 27 EXAMPLE 7 Using the Leontief Input-Output Model Solution continued Hence, the gross production matrix X is given by X I A D 1 1 36 16 1000 19 24 36 3000 84, 000 1 84, 000 19 19 132, 000 132, 000 19 © 2010 Pearson Education, Inc. All rights reserved 28 EXAMPLE 7 Using the Leontief Input-Output Model Solution continued Thus, to meet the consumer demand for 1000 units of energy and 3000 units of food, the 84, 000 energy produced must be units and 19 132, 000 units. food production must be 19 © 2010 Pearson Education, Inc. All rights reserved 29