Row-equivalent

Definition

Two matrices are row-equivalent if one can be obtained from the other by a sequence of elementary row operations. These operations include row swapping, scaling rows, and adding multiples of rows to each other.

5 Must Know Facts For Your Next Test

1. Row-equivalent matrices have the same solutions for their corresponding systems of linear equations.
2. Elementary row operations do not change the row-equivalence of a matrix.
3. If two augmented matrices are row-equivalent, they represent the same linear system.
4. The row-reduced echelon form (RREF) of two row-equivalent matrices is identical.
5. Row equivalence is an equivalence relation, meaning it is reflexive, symmetric, and transitive.

Review Questions

• What does it mean for two matrices to be row-equivalent?
• How do elementary row operations affect the solution set of a system of linear equations?
• Why is the concept of row equivalence important when solving systems using Gaussian Elimination?

Related terms

Elementary Row Operations: Operations that include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of rows.

Gaussian Elimination: A method used to solve systems of linear equations by transforming the matrix into an upper triangular form using elementary row operations.

Row-Reduced Echelon Form (RREF): A unique form that a matrix can be reduced to using elementary row operations where each leading entry is $1$ and is the only non-zero entry in its column.