Elementary Row Operations
Our goal is to begin with an arbitrary matrix and apply operations that
respect row equivalence until we have a matrix in Reduced Row Echelon
Form (RREF). The three elementary row operations are:
•(Row Swap) Exchange any two rows.
•(Scalar Multiplication) Multiply any row by a constant.
•(Row Sum) Add a multiple of one row to another row.
Why do these preserve the linear system in question? Swapping rows is
just changing the order of the equations begin considered, which certainly
should not alter the solutions. Scalar multiplication is just multiplying the
equation by the same number on both sides, which does not change the
solution(s) of the equation. Likewise, if two equations share a common
solution, adding one to the other preserves the solution.
There is a very simple process for row reducing a matrix, working column
by column. This system is called Gauss-Jordan Elimination.
1. If all entries in a given column are zero, then the associated variable is
undetermined; make a note of the undetermined variable(s) and then
ignore all such columns.
2. Swap rows so that the first entry in the first column is non-zero.
3. Multiply the first row by λso that the pivot is 1.
4. Add multiples of the first row to each other row so that the first entry
of every other row is zero.
5. Now ignore the first row and first column and repeat steps 1-5 until
the matrix is in RREF.
Example
3x3= 9
x1+5x2−2x3= 2
1
3x1+2x2= 3
First we write the system as an augmented matrix:
1
