Row Addition

Definition

Row addition is a fundamental operation in Gaussian elimination, a method used to solve systems of linear equations. It involves adding or subtracting multiples of one row to another row in order to eliminate variables and transform the system into an equivalent one that is easier to solve.

5 Must Know Facts For Your Next Test

1. Row addition is used to create zeros in the matrix, which helps to isolate variables and solve the system of equations.
2. The goal of row addition is to transform the augmented matrix into an upper triangular form, where the system can be solved using back-substitution.
3. Row addition is performed by multiplying a row by a constant and adding it to another row, effectively eliminating a variable from the second row.
4. The order in which row additions are performed is crucial, as it can affect the final solution and the stability of the numerical computations.
5. Row addition is a reversible operation, meaning that the original system of equations can be recovered from the transformed system.

Review Questions

• Explain how row addition is used in the Gaussian elimination method to solve a system of linear equations.
  • In the Gaussian elimination method, row addition is used to transform the augmented matrix of a system of linear equations into an upper triangular form. This is done by systematically creating zeros in the matrix, which helps to isolate variables and solve the system. Row addition involves multiplying a row by a constant and adding it to another row, effectively eliminating a variable from the second row. By performing a series of row additions, the matrix is transformed into a form where the system can be solved using back-substitution, ultimately finding the values of the variables that satisfy the original system of equations.
• Describe the role of row addition in the process of creating an upper triangular matrix during Gaussian elimination.
  • The primary purpose of row addition in Gaussian elimination is to transform the augmented matrix into an upper triangular form. This is achieved by strategically performing row additions to create zeros in the matrix, starting from the top-left corner and working downward. By eliminating variables in the lower rows, the matrix is gradually transformed into an upper triangular form, where the system of equations can be easily solved using back-substitution. The order and sequence of the row additions are crucial, as they can affect the final solution and the numerical stability of the computations.
• Analyze how the reversibility of row addition in Gaussian elimination allows for the recovery of the original system of linear equations.
  • Row addition is a reversible operation in Gaussian elimination, meaning that the original system of linear equations can be recovered from the transformed system. This is because row addition involves multiplying a row by a constant and adding it to another row, which is a linear transformation. By keeping track of the row operations performed, the original augmented matrix can be reconstructed, and the original system of equations can be recovered. This reversibility is an important property of Gaussian elimination, as it ensures that the solution obtained from the transformed system is indeed a valid solution to the original system of linear equations.

Related terms

Gaussian Elimination:

A systematic procedure for solving a system of linear equations by transforming the system into an equivalent one that is easier to solve, often by creating a triangular matrix.

Elementary Row Operations:

The three basic operations that can be performed on the rows of a matrix: row addition, row scaling, and row swapping.

Augmented Matrix:

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