Transforming Square Matrices Into Reduced Row Echelon Form

A matrix is a collection of numbers (and potentially fractions) placed into rows and columns. Specifically, a square matrix is a matrix with the same number of rows and columns. Understanding how to transform a matrix into a form called reduced row echelon form is important as this process is necessary in some more advanced linear algebra exercises such as inverting a matrix or solving linear systems of equations.

A square matrix is in reduced row echelon form when all entries in the main diagonal (which begins in the top left and ends in the bottom right) have a value of 1, and all other entries have a value of 0. There is also an intermediate form, called row echelon form. In row echelon form, only the entries below the main diagonal need to be 0s, and the main diagonal values do not necessarily have to be 1s (but must not be 0).

Time until completion may vary based on skill and matrix size, but should take less than 10 minutes. The following are what you will need in order to complete this process:

· Pencil

· Paper

· Calculator (optional)

· An understanding of addition, subtraction, multiplication, division, and fractions

Step 1: Determining If the Matrix Can Be Reduced

Not all square matrices can be transformed into reduced row echelon form. These matrices are referred to as being “noninvertible”. A square matrix will be noninvertible if any of the three following conditions are true:

· One row is identical to, or a constant multiple of, another row.

· A row of only zeros exists.

· A column of only zeros exists.

An example of each of these can be seen in the image above. If the matrix you are using meets one or more of these conditions, then you will not be able to complete the following steps.

In this Instructable, row 1 will be called R1, row 2 will be called R2, and so on. R1 is always the top row, and all other rows are located in numerical order below R1. Similarly, column 1, which is the leftmost column, will be called C1 and all other columns will be in numerical order to the right of C1.

Step 2: Row Operations

There are three row operations which can be performed in order to transform the matrix. Each of these operations must be applied to every element in a row.

· The first of these operations is switching rows; the order of rows can be moved around freely.

· Rows can also be multiplied by any number (including fractions).

· Lastly, the values of one row can be added to or subtracted from another row. Only entries that are in the same columns get added or subtracted with each other. Also, the row whose values are being added or subtracted can be multiplied or divided by any number (including fractions).

There is no exact order for these operations, as the process is dependent on the matrix that you are working with, and even with the same matrix there are multiple paths to a solution. Additionally, it might be possible to reduce the matrix without using all of these operations. Steps 3 through 6 will describe when to use each of these operations.

Step 3: Using Row Switching

It is easiest to reach row echelon form first before attempting to reach reduced row echelon form, which is why for now it is best to only focus on entries below the main diagonal. In the example above, the circled entries are the entries that need to be 0s in order for row echelon form to be achieved.

Always start by switching rows around so that as many 0s occupy entries below the main diagonal. In the example, R1 is switched with R3 so that a 0 occupies an entry below the main diagonal. It’s worth noting that not all matrices have conveniently placed 0s as this matrix does. If there are no rows with 0s that can be moved into position for row echelon form, then this step can be skipped.

When doing any row operation, it is important to show your work. Write down what operation you are doing, and then rewrite the matrix after performing that operation.

Step 4: Using Row Addition

As before, focus on changing entries below the main diagonal. Typically it is easiest to start with creating 0s in C1 by adding or subtracting with R1’s values. Next, focus on C2 by adding or subtracting R2’s values, and so on. In other words, start at the leftmost column, then move right.

Sometimes it is necessary to multiply the values being added or subtracted. For this example, R2 needs to be subtracted by 1/2 times R1 (R2 - (1/2)*R1) in order for the entry in both R2 and C1 to be turned into 0. Also, R3 can be subtracted by 2 times R2 (R3 - 2*R2), which will put this matrix in row echelon form.

In most matrices, more than one row addition will be necessary in order to reach row echelon form. Also, it is important to be careful when dealing with negative numbers as it is easy to mess up the signs.

Step 5: Using Row Multiplication

Once row echelon form is achieved, row multiplication should be used in order to change the values in the main diagonal into 1s. In the example, R1 must be divided by 2 (R1 / 2, which is equivalent to R1 * 1/2) and R3 must be divided by -2 (R3 / -2 or R3 * -1/2). Remember to either multiply or divide every number in a row whenever you perform row multiplication.

Step 6: Reduced Row Echelon Form

All that’s left is to transform the entries above the main diagonal into 0s. Start with the rightmost column, which in this matrix is C3. Use row addition with the bottom row, R3, in order to clear the entries in C3 that are above the main diagonal. Next, use row addition with R2 in order to clear the entries in C2 that are above the main diagonal. This will complete the process of transforming the matrix.

The process remains the same for matrices with more rows and columns than in this example. Start with the bottom row, and use it to clear entries in the rightmost column. Then, use the row above the bottom row to clear entries in the column next to the rightmost column. Continue this pattern until all entries above the main diagonal have been reduced to 0.

Step 7: More Practice

Above are some more matrices which can be transformed into reduced row echelon form. Additionally, you can create your own matrices with random values to use for practice. Don’t forget to check if it is possible for them to be reduced using step 1!

This process may seem trivial at first because the result is always the same. However, as mentioned in the introduction, understanding the process of using row operations to reduce a matrix is necessary for other linear algebra exercises such as matrix inversion and solving linear systems.

This website shows step by step solutions to reducing matrices. The order of steps used may differ from the order used in this Instructable: http://www.math.odu.edu/~bogacki/cgi-bin/lat.cgi?c=rref

This website shows how to use this process to invert a matrix: http://www.purplemath.com/modules/mtrxinvr.htm