Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
Elementary row operations are operations that can be performed on the rows of a matrix, including swapping rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another row
Elementary row operations are operations that can be performed on the rows of a matrix, including swapping rows, multiplying a row by a nonzero constant, and adding a multiple of one row to another row. An augmented matrix is a matrix that represents a system of linear equations, with the coefficients of the variables in the left part and the constants on the right part.
The statement “Elementary row operations on an augmented matrix never change the solution set of the associated linear system” is known as the Row Equivalent Theorem. This theorem states that if two augmented matrices are row equivalent, they have the same solution set. In other words, performing elementary row operations on an augmented matrix does not change the solutions to the system of linear equations that the matrix represents.
To understand why this is true, let’s consider the effect of each elementary row operation on the linear system:
1. Swapping rows: Swapping two rows of an augmented matrix corresponds to swapping the corresponding equations in the linear system. This does not change the solutions to the system since it is simply a rearrangement of the equations.
2. Multiplying a row by a nonzero constant: Multiplying a row of an augmented matrix by a nonzero constant is equivalent to multiplying the corresponding equation in the linear system by the same constant. This does not change the solutions to the system since multiplying both sides of an equation by a nonzero constant does not change the equality.
3. Adding a multiple of one row to another row: Adding a multiple of one row to another row in an augmented matrix corresponds to adding a multiple of the corresponding equation to another equation in the linear system. This operation does not change the solutions to the system because it does not change the relationships between the variables.
By performing these elementary row operations, we can manipulate the augmented matrix to put it in a desired form, such as row echelon form or reduced row echelon form. Despite the transformations, the solution set of the associated linear system remains unchanged.
In summary, the Row Equivalent Theorem guarantees that when we perform elementary row operations on an augmented matrix, the resulting matrix represents the same system of linear equations with the same solution set.
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