Is the statement Elementary row operations on an augmented matrix never change the solution set of the associated linear system true or false? Explain.
True, because the elementary row operations replace a system with an equivalent system.
The statement Elementary row operations on an augmented matrix never change the solution set of the associated linear system is true.
Elementary row operations are manipulations that we can perform on a matrix that do not change its row space or its null space. When we perform such operations on the augmented matrix of a linear system, we do not alter the set of solutions to the system. This is because the elementary row operations correspond to multiplying the system’s coefficient matrix by an invertible matrix. Multiplying the coefficient matrix by an invertible matrix does not change the solution set of the associated linear system.
Furthermore, elementary row operations can be used to bring the augmented matrix of a linear system into row echelon form or reduced row echelon form. The solution set of the system is then easily read off from the augmented matrix in this form. Since the solution set does not change when we perform elementary row operations, it follows that the solution set is preserved under the process of putting the augmented matrix into row echelon or reduced row echelon form.
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