8.7 Cramer’s Rule

INTRODUCTION

We saw at the end of the preceding section that a system of n linear equations in n variables AX = B has precisely one solution when det A ≠ 0. This solution, as we shall now see, can be expressed in terms of determinants. For example, the system of two equations in two variables

a11x1 + a12x2 = b1

a21x1 + a22x2 = b2 (1)

possesses the solution

(2)

provided that a11a22 − a12a21 ≠ 0. The numerators and denominators in (2) are recognized as determinants. That is, the system (1) has the unique solution

(3)

provided that the determinant of the matrix of coefficients ≠ 0. In this section we generalize the result in (3).

Using Determinants to Solve Systems

For a system of n linear equations in n variables

(4)

an1x1 + an2x2 + . . . + annxn = bn

it is convenient to define a special matrix

(5)

In other words, Ak is the same as the matrix A except that the kth column of A has been replaced by the entries of the column matrix

The generalization of (3), known as Cramer’s rule, is given in the next theorem. The mathematician Gabriel Cramer (1704–1752), born in the Republic of Geneva, published this method of solving linear systems using determinants in 1750.

THEOREM 8.7.1 Cramer’s Rule

Let A be the coefficient matrix of system (4). If det A ≠ 0, then the solution of (4) is given by

(6)

where Ak, k = 1, 2, . . . , n, is defined in (5).

PROOF:

We first write system (4) as AX = B. Since det A ≠ 0, A−1 exists, and so

Now the entry in the kth row of the last matrix is

(7)

But b1C1k + b2C2k + . . . + bnCnk is the cofactor expansion of det Ak, where Ak is the matrix given in (5), along the kth column. Hence, we have xk = det Ak/det A for k = 1, 2, . . . , n. ≡

EXAMPLE 1 Using Cramer’s Rule to Solve a System

Use Cramer’s rule to solve the system

SOLUTION

The solution requires the evaluation of four determinants:

Thus, (6) gives

≡

REMARKS

Like the method of the preceding section, Cramer’s rule is not a very practical means for solving systems of n linear equations in n variables. For n ≥ 4, the work required to evaluate the determinants becomes formidable. Nonetheless, as we saw in Section 3.5, Cramer’s rule is used sometimes, and it is of theoretical importance.

In the application of Cramer’s rule, some shortcuts can be taken. In Example 1, for instance, we really did not have to compute det A3 since once we found the values of x1 and x2, the value of x3 can be found by using one of the equations in the system.

8.7 Exercises Answers to selected odd-numbered problems begin on page ANS-19.

In Problems 1−10, solve the given system of equations by Cramer’s rule.



















  11. Use Cramer’s rule to determine the solution of the system

    For what value(s) of k is the system inconsistent?

  12. Consider the system

    When ϵ is close to 1, the lines that make up the system are almost parallel.

    1. Use Cramer’s rule to show that a solution of the system is
    2. The system is said to be ill-conditioned since small changes in the input data (for example, the coefficients) causes a significant or large change in the output or solution. Verify this by finding the solution of the system for ϵ = 1.01 and then for ϵ = 0.99.
  13. The magnitudes T1 and T2 of the tensions in the support wires shown in FIGURE 8.7.1 satisfy the equations

    Use Cramer’s rule to solve for T1 and T2.

    Two support wires hold a mass of 300 pounds placed equidistantly from the two sides. The support wire on the right is labeled T subscript 1 and is pulling the mass at an angle of 25 degrees. The support wire on the left is labeled T subscript 2 and is pulling the mass at an angle of 15 degrees.

    FIGURE 8.7.1 Support wires in Problem 13

  14. The 400-lb block shown in FIGURE 8.7.2 is kept from sliding down the inclined plane by friction and a force of smallest magnitude F. If the coefficient of friction between the block and the inclined plane is 0.5, then the magnitude of the frictional force is 0.5N, where N is the magnitude of the normal force exerted on the block by the plane. Use the fact that the system is in equilibrium to set up a system of equations for F and N. Use Cramer’s rule to solve for F and N.
    A 400 pound block is placed on an inclined plane. The angle of inclination from the horizontal line is 30 degrees. Four arrows point at the block. Arrow 1: An upward arrow to the right labeled F parallel to the inclination of the plane. Arrow 2: A vertical downward arrow labeled 400 lb. Arrow 3: An upward arrow to the left labeled N, perpendicular to the plane forming an angle of 60 degrees from the horizontal. Arrow 4: A small upward arrow to the right labeled 0.5 N along the line of the plane.

    FIGURE 8.7.2 Inclined plane in Problem 14

  15. As shown in FIGURE 8.7.3, a circuit consists of two batteries with internal resistances r1 and r2 connected in parallel with a resistor. Use Cramer’s rule to show that the current i through the resistor is given by

    An electric circuit consists of two batteries E subscript 1 and E subscript 2 with internal resistances r subscript 1 and r subscript 2 in parallel with a resistor R.

    FIGURE 8.7.3 Circuit in Problem 15