MATH 105: Finite Mathematics 2-6: The Inverse of a Matrix

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1 MATH 05: Finite Mathematics 2-6: The Inverse of a Matrix Prof. Jonathan Duncan Walla Walla College Winter Quarter, 2006

2 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of Equations 4 Conclusion

3 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of Equations 4 Conclusion

4 Solving Equations Recall that last time we saw that a system of equations can be represented as a matrix equation as shown below. Example Write the following system of equations in matrix form.

5 Solving Equations Recall that last time we saw that a system of equations can be represented as a matrix equation as shown below. Example Write the following system of equations in matrix form. { 2x + 3y = 7 3x 4y = 2

6 Solving Equations Recall that last time we saw that a system of equations can be represented as a matrix equation as shown below. Example Write the following system of equations in matrix form. { 2x + 3y = 7 3x 4y = 2 [ ] [ ] 2 3 x = 3 4 y [ ] 7 2

7 Solving Equations Recall that last time we saw that a system of equations can be represented as a matrix equation as shown below. Example Write the following system of equations in matrix form. { 2x + 3y = 7 3x 4y = 2 [ ] [ ] 2 3 x = 3 4 y [ ] 7 2 AX = B

8 Solving Equations Recall that last time we saw that a system of equations can be represented as a matrix equation as shown below. Example Write the following system of equations in matrix form. { 2x + 3y = 7 3x 4y = 2 [ ] [ ] 2 3 x = 3 4 y [ ] 7 2 AX = B If we wish to use the matrix equation on the right to solve a system of equations, then we need to review how we solve basic equations involving numbers.

9 Solving a Simple Equation The most basic algebra equation, ax = b, is solved using the multiplicative inverse of a. Example Solve the equation 3x = 6 for x. Step : Multiply by 3 Step 2: Simplify the Right 3 ( (3x) = 3 (6) 3 3) x = 2 Step 3: Simplify the Left x = 2 Step 4: Solution x = 2

10 Solving a Simple Equation The most basic algebra equation, ax = b, is solved using the multiplicative inverse of a. Example Solve the equation 3x = 6 for x. Step : Multiply by 3 Step 2: Simplify the Right 3 ( (3x) = 3 (6) 3 3) x = 2 Step 3: Simplify the Left x = 2 Step 4: Solution x = 2

11 Solving a Simple Equation The most basic algebra equation, ax = b, is solved using the multiplicative inverse of a. Example Solve the equation 3x = 6 for x. Step : Multiply by 3 Step 2: Simplify the Right 3 ( (3x) = 3 (6) 3 3) x = 2 Step 3: Simplify the Left x = 2 Step 4: Solution x = 2

12 Solving a Simple Equation The most basic algebra equation, ax = b, is solved using the multiplicative inverse of a. Example Solve the equation 3x = 6 for x. Step : Multiply by 3 Step 2: Simplify the Right 3 ( (3x) = 3 (6) 3 3) x = 2 Step 3: Simplify the Left x = 2 Step 4: Solution x = 2

13 Solving a Simple Equation The most basic algebra equation, ax = b, is solved using the multiplicative inverse of a. Example Solve the equation 3x = 6 for x. Step : Multiply by 3 Step 2: Simplify the Right 3 ( (3x) = 3 (6) 3 3) x = 2 Step 3: Simplify the Left x = 2 Step 4: Solution x = 2

14 Solving a Simple Equation The most basic algebra equation, ax = b, is solved using the multiplicative inverse of a. Example Solve the equation 3x = 6 for x. Step : Multiply by 3 Step 2: Simplify the Right 3 ( (3x) = 3 (6) 3 3) x = 2 Step 3: Simplify the Left x = 2 Step 4: Solution x = 2

15 Solving a Simple Equation The most basic algebra equation, ax = b, is solved using the multiplicative inverse of a. Example Solve the equation 3x = 6 for x. Step : Multiply by 3 Step 2: Simplify the Right 3 ( (3x) = 3 (6) 3 3) x = 2 Step 3: Simplify the Left x = 2 Step 4: Solution x = 2 This solution process worked because 3 is the inverse of 3, so that 3 =, the identity for multiplication. 3

16 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of Equations 4 Conclusion

17 Matrix Inverse To solve the matrix equation AX = B we need to find a matrix which we can multiply by A to get the identity I n. Matrix Inverse Let A be an n n matrix. Then a matrix A is the inverse of A if AA = A A = I n. Caution: Just as with numbers, not every matrix will have an inverse!

18 Matrix Inverse To solve the matrix equation AX = B we need to find a matrix which we can multiply by A to get the identity I n. Matrix Inverse Let A be an n n matrix. Then a matrix A is the inverse of A if AA = A A = I n. Caution: Just as with numbers, not every matrix will have an inverse!

19 Matrix Inverse To solve the matrix equation AX = B we need to find a matrix which we can multiply by A to get the identity I n. Matrix Inverse Let A be an n n matrix. Then a matrix A is the inverse of A if AA = A A = I n. Caution: Just as with numbers, not every matrix will have an inverse!

20 Verifying Matrix Inverses Example [ ] [ ] 2 2 Show that and are inverses. 2 [ ] [ ] [ ] = = I [ ] [ ] [ ] = = I

21 Verifying Matrix Inverses Example [ ] [ ] 2 2 Show that and are inverses. 2 [ ] [ ] [ ] = = I [ ] [ ] [ ] = = I

22 Verifying Matrix Inverses Example [ ] [ ] 2 2 Show that and are inverses. 2 [ ] [ ] [ ] = = I [ ] [ ] [ ] = = I

23 Verifying Matrix Inverses Example [ ] [ ] 2 2 Show that and are inverses. 2 [ ] [ ] [ ] = = I [ ] [ ] [ ] = = I While it is relatively easy to verify that matrices are inverses, we really need to be able to find the inverse of a given matrix.

24 Finding a Matrix Inverse To find the inverse of a matrix A we will use the fact that AA = I n. Find A [ ] [ ] 3 2 Let A = and find A x x = 2. 4 x 3 x 4 [ ] [ ] [ ] AA 3 2 x x = I = 4 x 3 x 4 0 [ ] [ ] 3x + 2x 3 3x 2 + 2x 4 0 = x + 4x 3 x 2 + 4x 4 0 This gives two systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 { 3x2 + 2x 4 = 0 x 2 + 4x 4 =

25 Finding a Matrix Inverse To find the inverse of a matrix A we will use the fact that AA = I n. Find A [ ] [ ] 3 2 Let A = and find A x x = 2. 4 x 3 x 4 [ ] [ ] [ ] AA 3 2 x x = I = 4 x 3 x 4 0 [ ] [ ] 3x + 2x 3 3x 2 + 2x 4 0 = x + 4x 3 x 2 + 4x 4 0 This gives two systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 { 3x2 + 2x 4 = 0 x 2 + 4x 4 =

26 Finding a Matrix Inverse To find the inverse of a matrix A we will use the fact that AA = I n. Find A [ ] [ ] 3 2 Let A = and find A x x = 2. 4 x 3 x 4 [ ] [ ] [ ] AA 3 2 x x = I = 4 x 3 x 4 0 [ ] [ ] 3x + 2x 3 3x 2 + 2x 4 0 = x + 4x 3 x 2 + 4x 4 0 This gives two systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 { 3x2 + 2x 4 = 0 x 2 + 4x 4 =

27 Finding a Matrix Inverse To find the inverse of a matrix A we will use the fact that AA = I n. Find A [ ] [ ] 3 2 Let A = and find A x x = 2. 4 x 3 x 4 [ ] [ ] [ ] AA 3 2 x x = I = 4 x 3 x 4 0 [ ] [ ] 3x + 2x 3 3x 2 + 2x 4 0 = x + 4x 3 x 2 + 4x 4 0 This gives two systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 { 3x2 + 2x 4 = 0 x 2 + 4x 4 =

28 Finding a Matrix Inverse To find the inverse of a matrix A we will use the fact that AA = I n. Find A [ ] [ ] 3 2 Let A = and find A x x = 2. 4 x 3 x 4 [ ] [ ] [ ] AA 3 2 x x = I = 4 x 3 x 4 0 [ ] [ ] 3x + 2x 3 3x 2 + 2x 4 0 = x + 4x 3 x 2 + 4x 4 0 This gives two systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 { 3x2 + 2x 4 = 0 x 2 + 4x 4 =

29 Finding the Matrix Inverse, Cont... Finding a Matrix Inverse, Continued Solve the systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 [ 3 2 ] [ 0 4 ] 0 4 A = { 3x2 + 2x 4 = 0 x 2 + 4x 4 = [ ]

30 Finding the Matrix Inverse, Cont... Finding a Matrix Inverse, Continued Solve the systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 [ 3 2 ] [ 0 4 ] 0 4 A = { 3x2 + 2x 4 = 0 x 2 + 4x 4 = [ ]

31 Finding the Matrix Inverse, Cont... Finding a Matrix Inverse, Continued Solve the systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 [ 3 2 ] [ 0 4 ] 0 4 A = { 3x2 + 2x 4 = 0 x 2 + 4x 4 = [ ]

32 Finding the Matrix Inverse, Cont... Finding a Matrix Inverse, Continued Solve the systems of equations: { 3x + 2x 3 = x + 4x 3 = 0 [ 3 2 ] [ 0 4 ] 0 4 A = { 3x2 + 2x 4 = 0 x 2 + 4x 4 = [ ]

33 General Rule for Finding An Inverse Applying the lessons of the previous example yields a general procedure for finding the inverse of a matrix. Finding a Matrix Inverse To find the inverse of a n n matrix A, form the augmented matrix [A I n ] and use row reduction to transform it into [ I n A ]. Caution: It may not be possible to get I n on the left side of the matrix. If it is not possible, then the matrix A has no inverse.

34 General Rule for Finding An Inverse Applying the lessons of the previous example yields a general procedure for finding the inverse of a matrix. Finding a Matrix Inverse To find the inverse of a n n matrix A, form the augmented matrix [A I n ] and use row reduction to transform it into [ I n A ]. Caution: It may not be possible to get I n on the left side of the matrix. If it is not possible, then the matrix A has no inverse.

35 General Rule for Finding An Inverse Applying the lessons of the previous example yields a general procedure for finding the inverse of a matrix. Finding a Matrix Inverse To find the inverse of a n n matrix A, form the augmented matrix [A I n ] and use row reduction to transform it into [ I n A ]. Caution: It may not be possible to get I n on the left side of the matrix. If it is not possible, then the matrix A has no inverse.

36 Finding the Inverse of a 3 3 Matrix Example Find the inverse of the following matrix, if it exists

37 Finding the Inverse of a 3 3 Matrix Example Find the inverse of the following matrix, if it exists Row operations yield: And therefore there is no inverse.

38 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of Equations 4 Conclusion

39 Using A to Solve a System of Equations The main reason we are interested in matrix inverses is to solve a system of equations written in matrix form. Solving a System of Equations If A is the matrix of coefficients for a system of equations, X is the column vector containing the system variables, and B is the column vector containing the constants, then: AX = B A AX = A B I n X = A B X = A B Caution: A system of equations can only be solved in this way if it has a unique solution.

40 Using A to Solve a System of Equations The main reason we are interested in matrix inverses is to solve a system of equations written in matrix form. Solving a System of Equations If A is the matrix of coefficients for a system of equations, X is the column vector containing the system variables, and B is the column vector containing the constants, then: AX = B A AX = A B I n X = A B X = A B Caution: A system of equations can only be solved in this way if it has a unique solution.

41 Using A to Solve a System of Equations The main reason we are interested in matrix inverses is to solve a system of equations written in matrix form. Solving a System of Equations If A is the matrix of coefficients for a system of equations, X is the column vector containing the system variables, and B is the column vector containing the constants, then: AX = B A AX = A B I n X = A B X = A B Caution: A system of equations can only be solved in this way if it has a unique solution.

42 An Example Example Use a matrix equation to set-up and solve each system of equations given below. { x 2y = 3x + 4y = 3 2 x + y z = 6 3x y = 8 2x 3y + 4z = 3

43 An Example Example Use a matrix equation to set-up and solve each system of equations given below. { x 2y = 3x + 4y = 3 2 x + y z = 6 3x y = 8 2x 3y + 4z = 3

44 An Example Example Use a matrix equation to set-up and solve each system of equations given below. { x 2y = 3x + 4y = 3 2 x + y z = 6 3x y = 8 2x 3y + 4z = 3

45 Reusing Results One major advantage of solving a system of equations using a matrix equation is that if your matrix of coefficients A stays the same, but your matrix B changes, you can reuse most of your work. Example Solve each of the following systems of equations using the results from the last part of the previous question. x + y z = 2 3x y = 2x 3y + 4z = 0 2 x + y z = 0 3x y = 4 2x 3y + 4z = 3

46 Reusing Results One major advantage of solving a system of equations using a matrix equation is that if your matrix of coefficients A stays the same, but your matrix B changes, you can reuse most of your work. Example Solve each of the following systems of equations using the results from the last part of the previous question. x + y z = 2 3x y = 2x 3y + 4z = 0 2 x + y z = 0 3x y = 4 2x 3y + 4z = 3

47 Reusing Results One major advantage of solving a system of equations using a matrix equation is that if your matrix of coefficients A stays the same, but your matrix B changes, you can reuse most of your work. Example Solve each of the following systems of equations using the results from the last part of the previous question. x + y z = 2 3x y = 2x 3y + 4z = 0 2 x + y z = 0 3x y = 4 2x 3y + 4z = 3

48 Reusing Results One major advantage of solving a system of equations using a matrix equation is that if your matrix of coefficients A stays the same, but your matrix B changes, you can reuse most of your work. Example Solve each of the following systems of equations using the results from the last part of the previous question. x + y z = 2 3x y = 2x 3y + 4z = 0 2 x + y z = 0 3x y = 4 2x 3y + 4z = 3

49 Outline Solving a Matrix Equation 2 The Inverse of a Matrix 3 Solving Systems of Equations 4 Conclusion

50 Important Concepts Things to Remember from Section 2-6 A A = AA = I n for an n n matrix A 2 Find A by reducing [A I n ] to [I n A] 3 Solving systems of equations with matrix equations.

51 Important Concepts Things to Remember from Section 2-6 A A = AA = I n for an n n matrix A 2 Find A by reducing [A I n ] to [I n A] 3 Solving systems of equations with matrix equations.

52 Important Concepts Things to Remember from Section 2-6 A A = AA = I n for an n n matrix A 2 Find A by reducing [A I n ] to [I n A] 3 Solving systems of equations with matrix equations.

53 Important Concepts Things to Remember from Section 2-6 A A = AA = I n for an n n matrix A 2 Find A by reducing [A I n ] to [I n A] 3 Solving systems of equations with matrix equations.

54 Next Time... Next time we will review for our third and final in class exam. It will be over the sections we covered from chapters and 2. For next time Review for Exam

55 Next Time... Next time we will review for our third and final in class exam. It will be over the sections we covered from chapters and 2. For next time Review for Exam

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