Final answer:
Row operations are used to reduce an augmented matrix to echelon form to determine the consistency of the system. A system is consistent if it does not result in contradictory rows during reduction, and solutions can be found through back-substitution if the system has unique solutions.
Step-by-step explanation:
To apply row operations to an augmented matrix and reduce the system to echelon form, we typically perform a series of steps that include row swapping, scaling rows by a non-zero number, and adding or subtracting multiples of one row from another. To determine if the system is consistent, we look for rows that correspond to contradictions, such as a row of zeros ending in a non-zero number, which would indicate an inconsistent system.
If we successfully reduce the system to echelon form without encountering a contradiction, the system is consistent, and we can find the solution by back-substitution if the system has a unique solution, or by identifying the parameters that lead to infinitely many solutions if the system is underdetermined.
For example, consider an augmented matrix for a system of equations:
1 2 3 | 9
0 4 5 | 10
0 0 1 | 3
This is already in echelon form. Since there are no rows that lead to contradictions, we can conclude that the system is consistent. The solutions would then be found by back-substituting from the bottom row to the top.