Leading Entry

Definition

The leading entry in a matrix is the first non-zero element in a row or column, which is used to perform row operations and solve systems of linear equations.

5 Must Know Facts For Your Next Test

1. The leading entry is used to perform row operations, such as row scaling, row swapping, and row addition, to transform a matrix into row echelon form or reduced row echelon form.
2. In the context of solving systems of linear equations using matrices, the leading entry is used to eliminate variables and isolate the variables in the system.
3. The leading entry is crucial for determining the rank of a matrix, which is the number of linearly independent rows or columns in the matrix.
4. The leading entry is also important for determining the number of free variables in a system of linear equations, as the number of free variables is equal to the number of columns in the matrix minus the rank of the matrix.
5. The leading entry is used to determine the pivot columns in a matrix, which are the columns that contain the leading entries in the row echelon form or reduced row echelon form of the matrix.

Review Questions

• Explain the role of the leading entry in the process of transforming a matrix into row echelon form.
  • The leading entry in a matrix is the first non-zero element in a row or column, and it is used to perform row operations to transform the matrix into row echelon form. By using the leading entry as a pivot, the matrix can be transformed through row scaling, row swapping, and row addition to eliminate variables and isolate the variables in the system of linear equations represented by the matrix.
• Describe how the leading entry is used to determine the rank of a matrix and the number of free variables in a system of linear equations.
  • The leading entries in a matrix are crucial for determining the rank of the matrix, which is the number of linearly independent rows or columns in the matrix. The number of leading entries in the row echelon form or reduced row echelon form of the matrix is equal to the rank of the matrix. Additionally, the number of free variables in a system of linear equations is equal to the number of columns in the matrix minus the rank of the matrix, which can be determined by the number of leading entries.
• Analyze the significance of the leading entry in the context of solving systems of linear equations using matrices.
  • In the context of solving systems of linear equations using matrices, the leading entry is essential for performing the necessary row operations to transform the matrix into row echelon form or reduced row echelon form. By using the leading entry as a pivot, the matrix can be manipulated to isolate the variables in the system, allowing for the determination of the solution. The leading entry is crucial for identifying the pivot columns in the matrix, which are the columns that contain the leading entries and are used to represent the variables in the solution.

Related terms

Row Echelon Form:

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

Reduced Row Echelon Form:

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

Elementary Row Operations:

The three basic row operations performed on a matrix to transform it into row echelon form or reduced row echelon form.