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 systems have the same solution set. Row equivalence is a relation between two matrices in which one matrix can be transformed into another matrix by performing a finite sequence of elementary row operations. These operations include:
1. Swapping two rows
2. Multiplying a row by a scalar
3. Adding a multiple of one row to another row
Performing these operations on an augmented matrix does not change the solution set of the corresponding linear system.
Therefore, if two matrices representing two linear systems are row equivalent, they represent the same system of linear equations, and thus they have the same solution set. This means that the two systems have either a unique solution, no solution, or infinitely many solutions.
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