key term - Elementary Row Operations

Definition

Elementary row operations are fundamental transformations performed on the rows of a matrix to simplify the matrix and solve systems of linear equations. These operations preserve the solutions of the original system while making it easier to find the solution.

5 Must Know Facts For Your Next Test

1. The three elementary row operations are: row swapping, row scaling, and row addition.
2. Performing elementary row operations on a matrix does not change the solution set of the associated system of linear equations.
3. Row echelon form is a step towards finding the reduced row echelon form, which is the unique, simplest representation of a matrix.
4. Gaussian elimination uses elementary row operations to transform a matrix into row echelon form, making it easier to solve the system of linear equations.
5. The reduced row echelon form of a matrix can be used to determine the rank, null space, and solution set of the associated system of linear equations.

Review Questions

• Explain how elementary row operations can be used to solve systems of linear equations.
  • Elementary row operations, such as row swapping, row scaling, and row addition, can be used to transform the coefficient matrix of a system of linear equations into row echelon form. This simplifies the system and makes it easier to solve. By performing a sequence of these operations, the matrix can be transformed into reduced row echelon form, which provides the unique, simplest representation of the system and allows for the determination of the solution set, rank, and null space of the original system.
• Describe the relationship between elementary row operations and the solution set of a system of linear equations.
  • Performing elementary row operations on the coefficient matrix of a system of linear equations does not change the solution set of the original system. This is because the elementary row operations are reversible transformations that preserve the solutions. By transforming the matrix into row echelon form or reduced row echelon form, the solutions to the original system can be more easily identified, but the set of solutions remains the same. This property of elementary row operations is crucial for solving systems of linear equations using Gaussian elimination.
• Analyze the role of reduced row echelon form in the context of solving systems of linear equations using elementary row operations.
  • The reduced row echelon form of a matrix is the unique, simplest representation of the associated system of linear equations. By performing elementary row operations to transform a matrix into reduced row echelon form, the rank, null space, and solution set of the original system can be easily determined. The reduced row echelon form provides a clear and concise way to identify the variables that are free and those that are dependent, as well as the number of solutions. This information is essential for fully understanding and solving the system of linear equations using the techniques of Gaussian elimination and elementary row operations.

Related terms

Row Echelon Form:

A matrix is in row echelon form when the leading entry in each non-zero row is 1, and all other entries in that column are 0.

Reduced Row Echelon Form:

A matrix is in reduced row echelon form when it is in row echelon form and all leading entries are 1, and all other entries in that column are 0.

Gaussian Elimination:

A method of solving systems of linear equations by performing a sequence of elementary row operations to transform the coefficient matrix into row echelon form.