Practice problems for these concepts can be found at:

- Systems of Equations and Inequalities Solved Problems for Intermediate Algebra
- Systems of Equations and Inequalities Supplementary Problems for Intermediate Algebra

### Determinants

The process of solving systems of linear equations is rather cumbersome. Fortunately the procedure is repetitive in nature. We shall introduce a technique that facilitates the process through the use of determinants. A *determinant* is a number that is associated with a square array of numbers.

**Definition 1**. A2 × 2 *determinant*, designated by has the value *ad* – *bc*. It is called a *second–order determinant*.

The expression used to obtain the value of the determinant seems arbitrary. There are logical reasons for using the stated expression which we choose to omit at this point.

See solved problem 7.5.

A third–order determinant has three rows and three columns. Its value is defined as follows.

**Definition 2.** A3 × 3 determinant, designated by has value *a*_{1}*b*_{2}*c*_{3} + *b*_{1}*c*_{2}*a*_{3} + *c*_{1}*a*_{2}*b*_{3} – *a*_{3}*b*_{2}*c*_{1} – *b*_{3}*c*_{2}*a*_{1} – *c*_{3}*a*_{2}*b*_{1}.

Because this definition is very cumbersome and difficult to remember, we evaluate third–order determinants by a method called *expansion by minors*. The *minor of an element* (number or letter) of a 3 × 3 determinant is the 2 × 2 determinant that remains after you delete the row and column which contain the element. Therefore, to evaluate a 3 × 3 determinant by expansion by minors of elements in the first column, apply the following relationship.

Evaluate the 2 × 2 determinants as illustrated previously. Notice that the signs of the coefficients of the terms alternate. The result is unchanged if we expand the determinant by minors of elements in any column or row. The appropriate signs of the terms in the expansion are displayed by position in the array below.

See solved problems 7.6–7.8.

### Cramer's Rule

Cramer's rule may be employed to solve systems of linear equations. His rule expresses the solution for each variable as the quotient of two determinants. This allows us to readily employ computers to solve systems of linear equations. We first address systems of linear equations in two unknowns.

In practice, find *D* first since there is no unique solution if *D* = 0.

See solved problem 7.9.

Cramer's rule can be extended to 3 × 3 systems of linear equations.

See solved problem 7.10.

If *D* = 0, there is no unique solution to the system. The system is either inconsistent or dependent. If at least one, but not all, of *D*_{x}, *D*_{y}, or *D*_{z} as well as *D* is zero, the system is inconsistent. If *D*, *D*_{x}, *D*_{y}, and *D*_{z} are all zero, the system is dependent.

Refer to supplementary problem 7.6 to practice applying Cramer's rule.

Practice problems for these concepts can be found at:

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