Linear Equations: Solutions Using Determinants

The determinant of a 2 × 2 matrix is defined as follows:

The determinant of a 3 × 3 matrix can be defined as shown in the following.

Each minor determinant is obtained by crossing out the first column and one row.

Example 1: Evaluate the following determinant.

First find the minor determinants.

The solution is

To use determinants to solve a system of three equations with three variables (Cramer's rule), say x, y, and z, four determinants must be formed following this procedure:

1. Write all equations in standard form.

2. Create the denominator determinant, D, by using the coefficients of x, y, and z from the equations, and evaluate it .

3. Create the x-numerator determinant, D_x, the y-numerator determinant, D_y, and the z-numerator determinant, D_z, by replacing the respective x, y, and z coefficients with the constants from the equations in standard form and evaluate each determinant .

The answers for x, y, and z are

Example 2: Solve this system of equations, using Cramer's rule.

Find the minor determinants.

Use the constants to replace the x-coefficients.

Use the constants to replace the y-coefficients.

Use the constants to replace the z-coefficients.

Therefore,

The check is left to you. The solution is x = 1, y = –2, z = –3.

If the denominator determinant, D, has a value of zero, then the system is either inconsistent or dependent. The system is dependent if all the determinants have a value of zero. The system is inconsistent if at least one of the determinants, D_x, D_y, or D_z, has a value not equal to zero and the denominator determinant has a value of zero.

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