In the real number system, if a is a nonzero number, then there exists a number b such that ab = ba = 1. The number b is called the multiplicative inverse of the number a and is denoted by a⁻¹. For a square matrix A it is also important to know whether we can find another square matrix B of the same order such that AB = BA = I. We have the following definition.

For example, the matrix A = is nonsingular or invertible since the matrix B = is its inverse. To verify this, observe that

and

Unlike the real number system, where every nonzero number a has a multiplicative inverse, not every nonzero n × n matrix A has an inverse.

EXAMPLE 1 Matrix with No Inverse

An n × n matrix that has no inverse is called singular. If A is nonsingular, its inverse is denoted by B = A⁻¹.

Note that the symbol −1 in the notation A⁻¹ is not an exponent; in other words, A⁻¹ is not a reciprocal. Also, if A is nonsingular, its inverse is unique.

Properties

The following theorem lists some properties of the inverse of a matrix.

PROOF of (i):

This part of the theorem states that if A is nonsingular, then its inverse A⁻¹ is also nonsingular and its inverse is A. To prove that A⁻¹ is nonsingular, we have to show that a matrix B can be found such that A⁻¹B = BA⁻¹ = I. But since A is assumed to be nonsingular, we know from (1) that AA⁻¹ = A⁻¹A = I and, equivalently, A⁻¹A = AA⁻¹ = I. The last matrix equation indicates that the required matrix, the inverse of A⁻¹, is B = A. Consequently, (A⁻¹)⁻¹ = A. ≡

Theorem 8.6.1(ii) extends to any finite number of nonsingular matrices:

that is, the inverse of a product of nonsingular matrices is the product of the inverses in reverse order.

In the discussion that follows we are going to consider two different ways of finding A⁻¹ for a nonsingular matrix A. The first method utilizes determinants, whereas the second employs the elementary row operations introduced in Section 8.2.

Adjoint Method

Recall from (6) of Section 8.4 that the cofactor C_ij of the entry a_ij of an n × n matrix A is C_ij = (−1)^i+jM_ij, where M_ij is the minor of a_ij; that is, the determinant of the (n − 1) × (n − 1) submatrix obtained by deleting the ith row and the jth column of A.

The next theorem will give a compact formula for the inverse of a nonsingular matrix in terms of the adjoint of the matrix. However, because of the determinants involved, this method becomes unwieldy for n ≥ 4.

PROOF:

For brevity we prove the case when n = 3. Observe that

(3)

since det A = a_i1C_i1 + a_i2C_i2 + a_i3C_i3, for i = 1, 2, 3, are the expansions of det A by cofactors along the first, second, and third rows, and

in view of Theorem 8.5.9. Thus, (3) is the same as

A (adj A) = (det A) = (det A) I

or A((1/det A) adj A) = I. Similarly, it can be shown in exactly the same manner that ((1/det A) adj A)A = I. Hence, by definition, A⁻¹ = (1/det A) adj A. ≡

For future reference we observe in the case of a 2 × 2 nonsingular matrix

A =

that the cofactors are C₁₁ = a₂₂, C₁₂ = −a₂₁, C₂₁ = −a₁₂, and C₂₂ = a₁₁. In this case,

adj A=

It follows from (2) that

(4)

For a 3 × 3 nonsingular matrix

A =

and so on. After the adjoint of A is formed, (2) gives

(5)

EXAMPLE 2 Inverse of a Matrix

EXAMPLE 3 Inverse of a Matrix

Find the inverse of A = .

SOLUTION

Since det A = 12, we can find A⁻¹ from (5). The cofactors corresponding to the entries in A are

From (5) we then obtain

The reader is urged to verify that AA⁻¹ = A⁻¹A = I. ≡

We are now in a position to prove a necessary and sufficient condition for an n × n matrix A to have an inverse.

PROOF:

We shall first prove the sufficiency. Assume det A ≠ 0. Then A is nonsingular, since A⁻¹ can be found from Theorem 8.6.2.

To prove the necessity, we must assume that A is nonsingular and prove that det A ≠ 0. Now from Theorem 8.5.6, AA⁻¹ = A⁻¹A = I implies

(det A)(det A⁻¹) = (det A⁻¹)(det A) = det I.

But since det I = 1 (why?), the product (det A)(det A⁻¹) = 1 ≠ 0 shows that we must have det A ≠ 0. ≡

For emphasis we restate Theorem 8.6.3 in an alternative manner:

An n × n matrix A is singular if and only if det A = 0. (6)

EXAMPLE 4 Using (6)

Because of the number of determinants that must be evaluated, the foregoing method for calculating the inverse of a matrix is tedious when the order of the matrix is large. In the case of 3 × 3 or larger matrices, the next method is a particularly efficient means for finding A⁻¹.

Row Operations Method

Although it would take us beyond the scope of this book to prove it, we shall nonetheless use the following results:

It is convenient to carry out these row operations on A and I simultaneously by means of an n × 2n matrix obtained by augmenting A with the identity I as shown here:

The procedure for finding A⁻¹ is outlined in the following diagram:

EXAMPLE 5 Inverse by Elementary Row Operations

If row reduction of (A|I) leads to the situation

where the matrix B contains a row of zeros, then necessarily A is singular. Since further reduction of B always yields another matrix with a row of zeros, we can never transform A into I.

EXAMPLE 6 A Singular Matrix

A system of m linear equations in n variables x₁, x₂, . . ., x_n,

(7)

can be written compactly as a matrix equation AX = B, where

Special Case

Let us suppose now that m = n in (7) so that the coefficient matrix A is n × n. In particular, if A is nonsingular, then the system AX = B can be solved by multiplying both of the equations by A⁻¹. From A⁻¹(AX) = A⁻¹B, we get (A⁻¹A)X = A⁻¹B. Since A⁻¹A = I and IX = X, we have

(8)

EXAMPLE 7 Using (8) to Solve a System

EXAMPLE 8 Using (8) to Solve a System

Use the inverse of the coefficient matrix to solve the system

SOLUTION

We found the inverse of the coefficient matrix

A =

in Example 5. Thus, (8) gives

Consequently, x₁ = 19, x₂ = 62, and x₃ = −36. ≡

Uniqueness

When det A ≠ 0 the solution of the system AX = B is unique. Suppose not—that is, suppose det A ≠ 0 and X₁ and X₂ are two different solution vectors. Then AX₁ = B and AX₂ = B imply AX₁ = AX₂. Since A is nonsingular, A⁻¹ exists, and so A⁻¹(AX₁) = A⁻¹(AX₂) and (A⁻¹A)X₁ = (A⁻¹A)X₂. This gives IX₁ = IX₂ or X₁ = X₂, which contradicts our assumption that X₁ and X₂ were different solution vectors.

Homogeneous Systems

A homogeneous system of equations can be written AX = 0. Recall that a homogeneous system always possesses the trivial solution X = 0 and possibly an infinite number of solutions. In the next theorem we shall see that a homogeneous system of n equations in n variables possesses only the trivial solution when A is nonsingular.

PROOF:

We prove the sufficiency part of the theorem. Suppose A is nonsingular. Then by (8), we have the unique solution X = A⁻¹0 = 0. ≡

The next theorem will answer the question: When does a homogeneous system of n linear equations in n variables possess a nontrivial solution? Bear in mind that if a homogeneous system has one nontrivial solution, it must have an infinite number of solutions.

In view of Theorem 8.6.6, we can conclude that a homogeneous system of n linear equations in n variables AX = 0 possesses

• only the trivial solution if and only if det A ≠ 0, and
• a nontrivial solution if and only if det A = 0.

The last result will be put to use in Section 8.8.

8.6 Exercises Answers to selected odd-numbered problems begin on page ANS-18.

8.6.1 Finding the Inverse

In Problems 1 and 2, verify that the matrix B is the inverse of the matrix A.

In Problems 3–14, use Theorem 8.6.3 to determine whether the given matrix is singular or nonsingular. If it is nonsingular, use Theorem 8.6.2 to find the inverse.

In Problems 15−26, use Theorem 8.6.4 to find the inverse of the given matrix or show that no inverse exists.

In Problems 27 and 28, use the given matrices to find (AB)⁻¹.

27. 
28. 
29. If A⁻¹ = , what is A?
30. If A is nonsingular, then (A^T)⁻¹ = (A⁻¹)^T. Verify this for A = .
31. Find a value of x such that the matrix A = is its own inverse.
32. The rotation matrix

    M =

    played an important role in Problems 45–48 of Exercises 8.1. Find In the context of those earlier problems, what does represent?

33. A nonsingular matrix A is said to be orthogonal if A⁻¹ = A^T.

    (a) Verify that the matrix in Problem 32 is orthogonal.

    (b) Verify that A = is an orthogonal matrix.

34. Show that if A is an orthogonal matrix (see Problem 33), then det A = ±1.
35. Suppose A and B are nonsingular matrices. Then show that AB is nonsingular.
36. Suppose A and B are matrices and that either A or B is singular. Then show that AB is singular.
37. Suppose A is a nonsingular matrix. Then show that
38. Suppose . Then show that either or A is singular.
39. Suppose A and B are matrices, A is nonsingular, and . Then show that .
40. Suppose A and B are matrices, A is nonsingular, and . Then show that .
41. Suppose A and B are nonsingular matrices. Is necessarily nonsingular?
42. Suppose A is a nonsingular matrix. Then show that is nonsingular.
43. Suppose A and B are nonzero matrices and Then show that both A and B are singular.
44. Consider the 3 × 3 diagonal matrix

    A = .

    Determine conditions such that A is nonsingular. If A is nonsingular, find A⁻¹. Generalize your results to an n × n diagonal matrix.

8.6.2 Using the Inverse to Solve Systems

In Problems 45−52, use an inverse matrix to solve the given system of equations.

In Problems 53 and 54, write the system in the form AX = B. Use X = A⁻¹B to solve the system for each matrix B.

53. 7x₁ − 2x₂ = b₁

    3x₁ − 2x₂ = b₂,

    
    
    

54. 

    

In Problems 55–58, without solving, determine whether the given homogeneous system of equations has only the trivial solution or a nontrivial solution.

55. 
    
    
56. 
    
    
57. 
    
    
58. 
59. The system of equations for the currents i₁, i₂, and i₃ in the network shown in FIGURE 8.6.1 is

    

    where R_k and E_k, k = 1, 2, 3, are constants.

    (a) Express the system as a matrix equation AX = B.

    (b) Show that the coefficient matrix A is nonsingular.

    (c) Use X = A⁻¹B to solve for the currents.

    

60. Consider the square plate shown in FIGURE 8.6.2, with the temperatures as indicated on each side. Under some circumstances it can be shown that the approximate temperatures u₁, u₂, u₃, and u₄ at the points P₁, P₂, P₃, and P₄, respectively, are given by

    

    (a) Show that the above system can be written as the matrix equation

    

    (b) Solve the system in part (a) by finding the inverse of the coefficient matrix.