Elementary Row Operation

Theorem

Let $\mathbf{Ax}=\mathbf{b}$ : Augmented matrix of a linear system

If any of the following operations are done to an augmented matrix:

1. Multiply an row through by a nonzero constant
2. Interchange two row
3. Add a constant times one row to another

Then the underlying linear system in the resulting augmented matrix has the same solution set as the original

Tip

The above operations do not alter the solution set, and may be used to produce a succession of increasingly simple systems.

Remark

Because resulting matrix from elementary row operations retain the same solution as the original, it may be used to simplify and solve a linear system. See Solving linear system with elementary row operations for demonstration.

Reversing elementary row operations

It should be evident if we let $B$ be the matrix that results from $A$ by performing one of the elementary row operations, then the matrix $A$ can be recovered from $B$ by performing the corresponding operation in the following table:

Operations on $I$ that produces $E$	Operations on $E$ that reproduces $I$
Multiply row $i$ by $c\ne 0$	Multiply row $i$ by $1/c$
Interchange rows $i$ and $j$	Interchange rows $i$ and $j$
Add $c$ times row $i$ to row $j$	Add $-c$ times row $i$ to row $j$