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 other rows.

5 Must Know Facts For Your Next Test

1. Row-equivalent matrices have the same solution set for their corresponding systems of linear equations.
2. Elementary row operations do not change the row equivalence class of a matrix.
3. If two augmented matrices are row-equivalent, their corresponding systems have the same solutions.
4. Gaussian elimination is a process that uses elementary row operations to transform a given matrix into its row-echelon form or reduced row-echelon form.
5. The concept of row equivalence is fundamental in solving systems of linear equations using matrix methods.

Review Questions

Related terms

Elementary Row Operations:

Operations including swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting multiples of one row to another.

Row Echelon Form:

A form of a matrix where all nonzero rows are above any rows of all zeros, and each leading coefficient (pivot) is to the right of the leading coefficient in the previous row.

Reduced Row Echelon Form:

\(RREF\) is an extension of Row Echelon Form where every leading coefficient is $1$ and is the only non-zero entry in its column.