Elementary row operations are a set of basic transformations performed on the rows of a matrix that do not change the solution to the system of linear equations represented by that matrix. These operations are fundamental in the process of Gaussian elimination, which is used to solve systems of linear equations.
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The three elementary row operations are: (1) Interchanging two rows, (2) Multiplying a row by a non-zero constant, and (3) Adding a multiple of one row to another row.
Performing elementary row operations on a matrix does not change the solution to the system of linear equations represented by that matrix.
Transforming a matrix into row echelon form using elementary row operations is a key step in the Gaussian elimination method for solving systems of linear equations.
The reduced row echelon form of a matrix is unique and can be obtained by performing additional elementary row operations on the row echelon form.
Elementary row operations are often represented using matrices, where the transformation is applied by multiplying the original matrix by an elementary matrix.
Review Questions
Explain how the three elementary row operations can be used to transform a matrix into row echelon form.
The three elementary row operations - interchanging rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another row - can be applied to a matrix to transform it into row echelon form. By performing these operations strategically, the leading entry in each row can be made 1, and the entries below it can be made 0, creating the characteristic stair-step pattern of a matrix in row echelon form. This process is a crucial step in the Gaussian elimination method for solving systems of linear equations.
Describe how the reduced row echelon form of a matrix is obtained from its row echelon form, and explain the significance of this transformation.
To obtain the reduced row echelon form of a matrix from its row echelon form, additional elementary row operations are performed. Specifically, the leading 1 in each row is made the only non-zero entry in its column by subtracting multiples of that row from the other rows. This results in a matrix where the leading entry in each non-zero row is 1, and all other entries in that column are 0. The reduced row echelon form is unique and provides the most compact representation of the solution to the system of linear equations, making it a valuable tool for further analysis and interpretation.
Analyze how the use of elementary row operations in the Gaussian elimination method ensures that the solution to the system of linear equations is preserved throughout the transformation process.
The key property of elementary row operations is that they do not change the solution to the system of linear equations represented by the matrix. By performing a series of these operations, such as interchanging rows, multiplying a row by a constant, or adding a multiple of one row to another, the original system of equations is transformed into an equivalent system with the same solutions. This preservation of the solution is crucial in the Gaussian elimination method, as it allows the matrix to be transformed into row echelon form or reduced row echelon form without altering the underlying system of equations. This ensures that the final solution obtained through back-substitution accurately represents the original system of linear equations.
A matrix is in reduced row echelon form if it is in row echelon form and the leading entry in each non-zero row is 1, and all other entries in that column are 0.
A method for solving systems of linear equations by performing elementary row operations to transform the augmented matrix into row echelon form, and then back-substituting to find the solution.