key term - Row Echelon Form

Definition

Row echelon form is a special arrangement of the rows in a matrix or system of linear equations where the leading entry in each non-zero row is 1, and is the only non-zero entry in its column. This structure allows for efficient solving of systems of equations using Gaussian elimination.

5 Must Know Facts For Your Next Test

1. The row echelon form of a matrix or system of equations has a unique solution if and only if the leading entry in each non-zero row is the only non-zero entry in its column.
2. Gaussian elimination is used to transform a system of linear equations into row echelon form, which makes it easier to solve for the unknown variables.
3. The row echelon form of a matrix reveals the rank of the matrix, which is the number of linearly independent rows or columns.
4. Elementary row operations, such as row scaling, row switching, and row addition, are used to transform a matrix into row echelon form.
5. The reduced row echelon form of a matrix is the most simplified form, where the leading entry in each non-zero row is 1, and all other entries in that column are 0.

Review Questions

• Explain the purpose and benefits of transforming a system of linear equations into row echelon form.
  • Transforming a system of linear equations into row echelon form serves several important purposes. First, it allows for the efficient solving of the system using Gaussian elimination, as the leading entry in each non-zero row becomes the only non-zero entry in its column. This simplifies the process of back-substitution and finding the values of the unknown variables. Additionally, the row echelon form reveals the rank of the matrix, which indicates the number of linearly independent equations and the existence of a unique solution. By reducing the system to row echelon form, it becomes easier to determine the number of solutions and identify any inconsistencies or redundancies in the original system of equations.
• Describe the three elementary row operations used to transform a matrix into row echelon form.
  • The three elementary row operations used to transform a matrix into row echelon form are: 1. Row scaling: Multiplying a row by a non-zero constant to make the leading entry 1. 2. Row switching: Interchanging two rows to move the leading entry to the desired position. 3. Row addition: Adding a multiple of one row to another row to eliminate entries below the leading entry. These operations can be performed systematically to progressively transform the matrix into row echelon form, with the leading entry in each non-zero row being the only non-zero entry in its column.
• Analyze the relationship between the row echelon form of a matrix and the rank of the matrix, and explain how this information can be used to determine the number of solutions to a system of linear equations.
  • The row echelon form of a matrix is closely related to the rank of the matrix, which is the number of linearly independent rows or columns. The number of non-zero rows in the row echelon form of a matrix is equal to the rank of the matrix. This information can be used to determine the number of solutions to a system of linear equations. If the rank of the coefficient matrix is equal to the number of variables in the system, then the system has a unique solution. If the rank is less than the number of variables, then the system has infinitely many solutions. If the rank is greater than the number of variables, then the system is inconsistent and has no solutions. By transforming the system into row echelon form, you can easily identify the rank of the coefficient matrix and use this information to determine the number and nature of the solutions to the system of linear equations.

Related terms

Gaussian Elimination:

A method for solving systems of linear equations by systematically transforming the coefficient matrix into row echelon form.

Augmented Matrix:

A matrix formed by combining the coefficient matrix and the constant terms of a system of linear equations, used in Gaussian elimination.

Elementary Row Operations:

The three basic operations that can be performed on the rows of a matrix to transform it into row echelon form: row scaling, row switching, and row addition.