Copyright © Cengage Learning. All rights reserved.

2SYSTEMS OF

LINEAR EQUATIONS AND

MATRICES

Read pp. 129-30. Stop at “Inverse of a Matrix” box

Copyright © Cengage Learning. All rights reserved.

2.6 The Inverse of a Square Matrix

3

The Inverse of a Square Matrix

4

The Inverse of a Square Matrix

If a is a nonzero real number, then there exists a unique real number a–1(that is, ) such that

The use of the (multiplicative) inverse of a real number enables us to solve algebraic equations of the form

ax = b (13)

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The Inverse of a Square Matrix

Multiplying both sides of (13) by a–1, we have

For example, since the inverse of 2 is 2–1 = , we can solve the equation

2x = 5

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The Inverse of a Square Matrix

By multiplying both sides of the equation by 2

–1 = , giving

2

–1(2x) = 2

–1 5

We can use a similar procedure to solve the matrix equation

AX = B

where A, X, and B are matrices of the proper sizes.

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The Inverse of a Square Matrix

To do this we need the matrix equivalent of the inverse of a real number. Such a matrix, whenever it exists, is called the inverse of a matrix.

Not every square matrix has an inverse. A square matrix that has an inverse is said to be nonsingular. A matrix that does not have an inverse is said to be singular.

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The Inverse of a Square Matrix

An example of a singular matrix is given by

If B had an inverse given by

where a, b, c, and d are some appropriate numbers, then by the definition of an inverse.

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The Inverse of a Square Matrix

we would have BB–1 = I; that is,

which implies that 0 = 1—an impossibility! This contradiction shows that B does not have an inverse.

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Solving Systems of Equations with Inverses

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Solving Systems of Equations with Inverses

We now show how the inverse of a matrix may be used to solve certain systems of linear equations in which the number of equations in the system is equal to the number of variables.

For simplicity, let’s illustrate the process for a system of three linear equations in three variables:

a11x1 + a12x2 + a13x3 = b1

a21x1 + a22x2 + a23x3 = b2

a31x1 + a32x2 + a33x3 = b3

(15)

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Solving Systems of Equations with Inverses

Let’s write

You should verify that System (15) of linear equations may be written in the form of the matrix equation

AX = B (16)

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Solving Systems of Equations with Inverses

If A is nonsingular, then the method of this section may be used to compute A–1. Next, multiplying both sides of Equation (16) by A–1 (on the left), we obtain

A–1AX = A–1B or IX = A–1B or X = A–1B

the desired solution to the problem. In the case of a system of n equations with n unknowns, we have the following more general result.

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Solving Systems of Equations with Inverses

In the case of a system of n equations with n unknowns, we have the following more general result.

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Example 4

Solve the following systems of linear equations:

a. 2x + y + z = 1

3x + 2y + z = 2

2x + y + 2z = –1

b. 2x + y + z = 2

3x + 2y + z = –3

2x + y + 2z = 1

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Example 4 – Solution

We may write the given systems of equations in the form

AX = B and AX = C

respectively, where

cont’d

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Example 4 – Solution

The inverse of the matrix A,

was found in Example 1.

Using this result, we find that the solution of the first system (a) is

cont’d

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Example 4 – Solution

or x = 2, y = –1, and z = –2.

cont’d

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Example 4 – Solution

The solution of the second system (b) is

X = A–1 C

or x = 8, y = –13, and z = –1.

cont’d

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Practice

p. 137 Self-Check Exercises #1-3