Properties of Row Operations

Elementary row operations are the fundamental tools for manipulating linear systems. Understanding their properties ensures that our solution methods preserve correctness while transforming matrices into simpler forms.

TheoremRow Equivalence Preserves Solutions

If matrix $B$ is obtained from matrix $A$ by a sequence of elementary row operations, then the linear systems $A\mathbf{x} = \mathbf{0}$ and $B\mathbf{x} = \mathbf{0}$ have exactly the same solution set. We say $A$ and $B$ are row equivalent, denoted $A \sim B$ .

More generally, for augmented matrices, $[A|\mathbf{b}] \sim [B|\mathbf{c}]$ implies that the systems $A\mathbf{x} = \mathbf{b}$ and $B\mathbf{x} = \mathbf{c}$ have the same solution set.

This theorem is the theoretical foundation for Gaussian elimination. Since each elementary row operation is reversible, row equivalence defines an equivalence relation on the set of matrices, partitioning them into equivalence classes based on their solution sets.

TheoremInvertibility of Row Operations

Each elementary row operation has an inverse operation of the same type:

Row switch $R_i \leftrightarrow R_j$ is self-inverse
Row multiplication $R_i \to cR_i$ has inverse $R_i \to \frac{1}{c}R_i$
Row addition $R_i \to R_i + cR_j$ has inverse $R_i \to R_i - cR_j$

Therefore, if $B$ is obtained from $A$ by row operations, then $A$ can be recovered from $B$ by the inverse operations applied in reverse order.

ExampleRow Operation Matrices

Each elementary row operation can be represented by multiplying on the left by an elementary matrix. For example, the operation $R_2 \to R_2 - 3R_1$ on a $3 \times 3$ matrix corresponds to left multiplication by:

$E = \begin{bmatrix} 1 & 0 & 0 \\ -3 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$

Then $EA$ equals the result of performing the row operation on $A$ . The inverse operation $R_2 \to R_2 + 3R_1$ corresponds to $E^{-1} = \begin{bmatrix} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$ .

Remark

The representation of row operations as elementary matrices establishes a profound connection: solving $A\mathbf{x} = \mathbf{b}$ by row operations is equivalent to finding elementary matrices $E_1, E_2, \ldots, E_k$ such that $E_k \cdots E_2 E_1 A = \text{RREF}$ . This perspective leads directly to the concept of matrix factorization and the LU decomposition.