The three elementary row operations on an augmented matrix
Row operations let us replace a linear system with an easier system that has the same solutions. Each operation changes how the equations are written. We’ll look at the three allowed operations and what each one does to an augmented matrix.
Swap two rows
The notation means exchange the entire first and second rows. For the system and , start with
After the swap, the matrix is
The equations appear in a different order, so their common solutions stay the same.
Multiply a row by a nonzero scalar
Using the original matrix again, doubles every entry of row 1, including its constant.
The first equation becomes . Dividing by 2 recovers the original equation, which is why the solution set is preserved. The scalar must be nonzero so the operation can be reversed.
Add a multiple of one row to another
In the original matrix, replaces only row 1. Adding corresponding entries gives
Row 2 stays unchanged. Subtracting that row from the new row 1 reverses the operation.
Work with a larger augmented matrix
Apply . Four times row 3 is , so the replacement first row is .
Apply combined changes one operation at a time
From the original larger matrix, multiply row 1 by 3 and row 3 by 2, then swap those two rows. The final rows are , , and . Separating the scaling and swap makes the destination of each row clear.