Reduction to Row Echelon Form
A matrix in row echelon form is one in which all the elements below the formed by the leading coefficients are zero and all the leading coefficients are ones. The leading coefficient of a matrix is referred to as the first non-zero element of a matrix. Contrary to popular belief, most non-square matrices can also be reduced to row echelon form.
Below are a few examples of matrices in row echelon form:
In all the matrices above, the first non-zero element of each row is one and hence all the elements below the diagonal formed by these leading coefficients are zero.
Reducing matrices to row echelon form makes them easier to work with when it comes to finding solutions to systems of equations.
Row Operations on Matrices
The rows and columns in a matrix can be manipulated in certain ways while at the same time maintaining the integrity of the matrix. Although the same operations apply to both rows and columns, we're going to focus on rows.
The are three fundamental row operations that can be performed on a matrix:
Row switching involves switching one row with another in a matrix. This will not change the matrix as long as the entire row is switched for another. It is important though to keep track of what rows you've switched.
In the above, the positions of rows Ri and Rj are switched. For example, let's switch the positions of row 1 ( R1) with row 3 ( R3):
results in the matrix below
The new names of the rows ( R1' and R3') indicate the rows have been modified in some way.
There are a number of reasons as to why you'd choose to switch the rows of your matrix, but the main reason is that certain rows in the matrix might be easier to work with than others.
Row multiplication involves multiplying a given row by a non-zero constant. This too doesn't change the meaning of the matrix as long as we keep track of the operation and multiply the entire row by the constant.
Below is an example of row multiplication in matrices
The operation is
where x is a non-zero constant. The above results in the following matrix
Row addition involves adding one row to another. This is operation is usually carried out after row multiplication and is usually performed with the intention of eliminating one of the elements from a row. Matrix addition takes the following form:
Adding Rows 1 and 2 (R2' = R1' + R2')
Row Operations and Row Echelon Form
Row operations are used to reduce a matrix ro row echelon form. Let's review a few examples to see how it all works.
Reduce the following matrix to row echelon form
The first step is to label the matrix rows so that we can know which row we're referring to.
The next step in reducing a matrix to row echelon is to make sure that the leading element in the first row is one. In order to achieve this in the above matrix, we switch rows 2 and 1.
The next step involves both row multiplication and addition. We need to get rid of all the elements below the leading element of the first row by making them zeros. To achieve this, we find a constant which when multiplied by the row and then the result added to the first row, the first element of the row becomes zero.
For R2 to get change the first element to zero, we need to multiply the entire row by a constant such that when row 2 is added to row 1, the first element of the new row 2 is zero.
Since first element is 4, we need to multiply it by -1⁄4 and then add the result to row 1.
Adding the result to row 1:
We do the same for row 3. Since the first element in row 3 is also four, we can just repeat the above procedure.
Adding the result to row 1:
Now we're much closer to row echelon form, but the leading coefficient in row 2 is not one so we need to change that. We can achieve that by multiplying row 2 by 4
We're left with one more row to change; row 3. We need to element below the leading coefficient of the previous row into a zero without tampering with the first element in row 3.
This rules out using row 1 to achieve our goal. But since we also have a zero in the first element of row 2, we can use row 2 to change row 3 without disturbing the first elements of either row.
We multiply row 3 by 4 and then add the result to row 2 as follows:
Adding the result
One more step: We need to change the leading coefficient of the last row into one, and we can achieve this by multiplying the row by 1⁄18
The matrix has hence been reduced to row echelon form: