11. Systems of Linear Equations: Cramer’s Rule

Section 1.11. Systems of Linear Equations: Cramer’s Rule

The formula (1.10.61), repeated,

is useful when one wants to calculate the inverse. While they are numerically more useful algorithms for large systems, (1.11.1) is a formula with a lot of applications. One of these arises

NN when one knows that a matrix  A M is nonsingular and the goal is to find the solution to the

linear system

A x  b (1.11.2)

We know from Section 1.6 that this system has a unique solution when the matrix A is nonsingular. Equation (1.11.1) allows us to write that solution in the explicit form

 adj A  b (1.11.3)

det A

Example 1.11.1: In the special case N  2 , we can use the formula (1.10.66) to express the

solution (1.11.3) as

 (1.11.4)  x 2 AA 11 22  AA 12 21   A 21  A 11  b 2   bA 1 21  bA 2 11 

 AA 11 22  AA 12  21 

One way the solution (1.11.4) is sometimes written is

Equation (1.11.5) is an example of Cramer’s Rule 7 . One simply places in the numerator of each formula the determinant formed from the determinant of the matrix of coefficients except that the

first column is replaced by the components of b in the formula for x 1 and the second column is replaced by the components of b in the formula for x 2 .

7 This rule is named after the Swiss mathematician Gabriel Cramer. Information about Gabriel Cramer can be found, for example, at http://en.wikipedia.org/wiki/Gabriel_Cramer .

118 Chap. 1 • ELEMENTARY MATRIX THEORY

Just as (1.11.5) follows from (1.11.3), Cramer’s rule for systems of arbitrary order also follow from (1.11.3). The first step is to express (1.11.3) in the component form

  cof Ab ij  i det (1.11.6)

Just as (1.10.53) expresses the determinant as a cofactor expansion, one can recognize the numerator of (1.11.6) as a determinant. Except that it is a determinant with the th j column of A

replaced by the components of b . For example, in the case N  4 , the four solutions are given by

Example 1.11.2: In Example 1.3.1, we solved the system (1.3.2), repeated,

2 x 1  x 2  x 3  3 (1.11.8)

 x 1 2 x 2  3 x 3  7

Cremer’s rule tells us that if det A  0 the solution is

Sec. 1.11 • Systems of Linear Equations: Cramer’s Rule

b 1 A 12 A 13 1 2  1 b 2 A 22 A 23 3  1 1

b 3 A 32 A 33 7 2 3 1   5 38  71  22

A 11 A 12 A 13 1 2  1 1   5 28  11  22

A 21 A 22 A 23 2  1 1 A 31 A 32 A 33  1 2 3

A 11 b 1 A 13 1 1  1 A 21 b 2 A 23 2 3 1

A 31 b 3 A 33  17 3 12   2 10  14  22

A 11 A 12 A 13 1 2  1 1   5 28  11   22

A 21 A 22 A 23 2  1 1 A 31 A 32 A 33  1 2 3

A 11 A 12 b 1 1 2 1 A 21 A 22 b 2 2  13

A 31 A 32 b 3  1 2 7 1   13   2 12  17  44

A 11 A 12 A 13 1 2  1 1   5 28  11  22

A 21 A 22 A 23 2  1 1 A 31 A 32 A 33  1 2 3

which, of course, is the earlier result (1.3.10).

Exercises

1.11.1: Utilize Cremer’s rule to find the solution of the system

2 x 1  4 x 2  9  (1.11.10)

1.11.2: Utilize Cremer’s rule to find the solution of the system

x 1  2 x 2  3 x 3  6 (1.11.11)

1.11.3: Utilize Cremer’s rule to find the solution of the system

120 Chap. 1 • ELEMENTARY MATRIX THEORY

3 x 1  x 2 2 x 3  6

2 x 1  x 2  x 3  6 (1.11.12)

1.11.4: Utilize Cremer’s rule to find the solution of the system introduced in Exercise 1.3.2, i.e., the system

2 x 1  x 2  x 3  5 (1.11.13)

 2 x 1  2 x 2  x 3  8

1.11.5: Utilize Cremer’s rule to find the solution of the system introduced in Exercise 1.3.3, i.e., the system

4 x 1  6 x 2  7 x 3  3 (1.11.14)

1.11.6: Utilize Cremer’s rule to show that the solution of the system

2 x 1  2 x 2  1 x 3  6 (1.11.15)

 1 is x  1 .

1.11.7: Utilize Cremer’s rule to show that the solution of the system

Sec. 1.11 • Systems of Linear Equations: Cramer’s Rule 121

is x    1

122 Chap. 1 • ELEMENTARY MATRIX THEORY

__________________________________________________________ Chapter 2