College Algebra Chapter 6 Matrices and Determinants and Applications Section 6.1 Solving Systems of Linear Equations Using Matrices

Concepts 1. Write an Augmented Matrix 2. Use Elementary Row Operations 3. Use Gaussian Elimination and Gauss-Jordan Elimination

Write an Augmented Matrix Matrix: A rectangular array of numbers Examples: A matrix can be used to represent a system of linear equations written in standard form. To do so, we extract the coefficients of each term in the equation to form an augmented matrix. A bar within the augmented matrix separates the coefficients of the variable terms in the equations from the constant terms.

Example 1: Write an augmented matrix for the system.

Example 2: Write an augmented matrix for the system.

Example 3: Write an augmented matrix for the system.

Example 4: Write a system of linear equations represented by the augmented matrix.

Example 5: Write a system of linear equations represented by the augmented matrix.

Concepts 1. Write an Augmented Matrix 2. Use Elementary Row Operations 3. Use Gaussian Elimination and Gauss-Jordan Elimination

Use Elementary Row Operations 1. Interchange two rows R1  R2 interchange rows 1 and row 2 2. Multiply a row by a nonzero constant 3R1  R1 multiply row 1 by 3 and then replace row 1 by the result

Use Elementary Row Operations 3. Add a multiple of one row to another row 2R1 + R2  R2 multiply row 1 by 2, add it to row 2, and then replace row 2 by the result

Perform the following elementary row operations. Example 6: Perform the following elementary row operations. 1 2 R1R1 R1 + R2 R2

Perform the following elementary row operations. Example 7: Perform the following elementary row operations. –R3 + R2  R2 − 1 9 R2R2

Concepts 1. Write an Augmented Matrix 2. Use Elementary Row Operations 3. Use Gaussian Elimination and Gauss-Jordan Elimination

Use Gaussian Elimination and Gauss-Jordan Elimination If we perform repeated row operations, we can form an augmented matrix that represents a system of equations that is easier to solve than the original system. In particular, it is easy to solve a system whose augmented matrix is in row-echelon form or reduced row-echelon form.

Use Gaussian Elimination and Gauss-Jordan Elimination Row-echelon form: 1. Any rows consisting entirely of zeros are at the bottom of the matrix. 2. For all other rows, the first nonzero entry is 1. This is called the leading 1. The leading 1 in each nonzero row is to the right of the leading 1 in the row immediately above. Examples:

Use Gaussian Elimination and Gauss-Jordan Elimination Reduced row-echelon form: The matrix is in row-echelon form with the added condition that each row with a leading entry of 1 has zeros above the leading 1. Examples:

Use Gaussian Elimination and Gauss-Jordan Elimination After writing an augmented matrix in row-echelon form, the corresponding system of linear equations can be solved using back substitution. This method is called Gaussian elimination.

Use Gaussian Elimination and Gauss-Jordan Elimination To solve a system of linear equations using Gaussian elimination: Step 1: Write the augmented matrix for the system.   Step 2: Use elementary row operations to write the augmented matrix in row-echelon form. (Begin by getting the entry in row 1 column 1 to be 1, with zeros under it.) Step 3: Use back substitution to solve the resulting system of equations.

Example 8: Solve the system using Gaussian elimination. Write the equations in standard form: Write the augmented matrix:

Example 8 continued: Perform the row operations:

Example 9: Solve the system using Gaussian elimination.

Example 9 continued:

Use Gaussian Elimination and Gauss-Jordan Elimination If we write an augmented matrix in reduced row-echelon form, we can solve the corresponding system of equations by inspection. This is called the Gauss-Jordan elimination method.

Example 10: Continue to perform row operations on the matrix in (9) until it is in reduced row-echelon form. Then determine the solution by inspection.

Example 10 continued:

Example 11: Solve the system by using Gauss-Jordan elimination.

Example 11 continued:

Example 12: University of Texas at Austin has three times as many students enrolled as University of Miami. University of California, Berkley has 3,000 more than twice the number of students as University of Miami. If the three schools have a total enrollment of 96,000 students, what is the enrollment at each school?

Example 12 continued:

Example 12 continued:

Example 12 continued: