Equivalent Systems

Equivalent systems are the systems with the same set of solutions.

2 linear systems are equivalent iff one is obtained from the other by a finite number of elementary operations.

Applying elementary operations to the augmented matrix $[A \mid \vec{b}]$ instead of the equations gives row equivalence (see Elementary Transformations).

Row-Equivalent Matrix

2 matrices are said to be row-equivalent iff one can be obtained from the other by a finite number of elementary row operations.