If the augmented matrices of two linear systems are row equivalent…
The 2 systems have the same solution set
If the augmented matrices of two linear systems are row equivalent, it means that the two systems have the same solution set. Row equivalence means that one can obtain one matrix from the other by performing a sequence of elementary row operations. In other words, the corresponding systems have the same solution set because they are equivalent systems.
Elementary row operations include:
1. Switching two rows.
2. Multiplying a row by a nonzero constant.
3. Adding a multiple of one row to another row.
Since these operations do not change the solution set of the linear system, it follows that if two linear systems have row equivalent augmented matrices, they have the same solution set. This result is known as the row equivalence theorem and can be used to simplify calculations when finding the solution of a linear system.
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