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, then the two systems have the same solutions.
Row equivalence means that the augmented matrices can be transformed from one to the other through a series of elementary row operations. These operations include swapping two rows, multiplying a row by a non-zero constant, and adding a multiple of one row to another row.
Since the elementary row operations do not change the solutions of the system, if the augmented matrices are row equivalent, the corresponding linear systems are equivalent. This means that they have the same solutions or satisfy the same set of equations.
Therefore, if you solve one of the systems, you will get the solution to the other system as well. Row equivalence is a useful tool when solving a system of linear equations using the Gaussian elimination method, as it allows us to simplify the system without changing its solutions.
More Answers:
Pivot Positions And Pivot Columns In Linear Algebra And Matrix TheoryRow Operations And Reduced Echelon Matrices: A Comprehensive Guide
Matrix Echelon And Reduced Echelon Form For Efficient Algebraic Operations