Systems of Linear Equations
Solve Systems of Equations Using Determinants
Learning Objectives
By the end of this section, you will be able to:
- Evaluate the determinant of a matrix
- Evaluate the determinant of a matrix
- Use Cramer’s Rule to solve systems of equations
- Solve applications using determinants
Before you get started, take this readiness quiz.
In this section we will learn of another method to solve systems of linear equations called Cramer’s rule. Before we can begin to use the rule, we need to learn some new definitions and notation.
Evaluate the Determinant of a Matrix
If a matrix has the same number of rows and columns, we call it a square matrix. Each square matrix has a real number associated with it called its determinant. To find the determinant of the square matrix we first write it as To get the real number value of the determinate we subtract the products of the diagonals, as shown.
The determinant of any square matrix where a, b, c, and d are real numbers, is
Evaluate the determinate of ⓐ ⓑ
ⓐ
| Write the determinant. | |
| Subtract the products of the diagonals. | |
| Simplify. | |
| Simplify. |
ⓑ
| Write the determinant. | |
| Subtract the products of the diagonals. | |
| Simplify. | |
| Simplify. |
Evaluate the determinate of ⓐ ⓑ
ⓐⓑ
Evaluate the determinate of ⓐ ⓑ
ⓐ 2 ⓑ
Evaluate the Determinant of a Matrix
To evaluate the determinant of a matrix, we have to be able to evaluate the minor of an entry in the determinant. The minor of an entry is the determinant found by eliminating the row and column in the determinant that contains the entry.
The minor of an entry in a determinant is the determinant found by eliminating the row and column in the determinant that contains the entry.
To find the minor of entry we eliminate the row and column which contain it. So we eliminate the first row and first column. Then we write the determinant that remains.
To find the minor of entry we eliminate the row and column that contain it. So we eliminate the 2nd row and 2nd column. Then we write the determinant that remains.
For the determinant find and then evaluate the minor of ⓐ ⓑ ⓒ
ⓐ
| Eliminate the row and column that contains | |
| Write the determinant that remains. | |
| Evaluate. | |
| Simplify. |
ⓑ
| Eliminate the row and column that contains | |
| Write the determinant that remains. | |
| Evaluate. | |
| Simplify. |
ⓒ
| Eliminate the row and column that contains | |
| Write the determinant that remains. | |
| Evaluate. | |
| Simplify. |
For the determinant find and then evaluate the minor of ⓐ ⓑ ⓒ
ⓐ 3 ⓑ 11 ⓒ 2
For the determinant find and then evaluate the minor of ⓐ ⓑ ⓒ
ⓐⓑ 2 ⓒ 3
We are now ready to evaluate a determinant. To do this we expand by minors, which allows us to evaluate the determinant using determinants—which we already know how to evaluate!
To evaluate a determinant by expanding by minors along the first row, we use the following pattern:
Remember, to find the minor of an entry we eliminate the row and column th