Get Reduced: 5 Steps To Row Reduced Echelon Form

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The world of linear algebra can be a daunting place, especially when dealing with matrices. One of the most powerful tools in this field is the Reduced Row Echelon Form (RREF). In this article, we'll break down the 5 steps to achieve RREF and make your linear algebra journey smoother.

What is Reduced Row Echelon Form?

Before we dive into the steps, let's quickly define what RREF is. RREF is a specific format for matrices that makes it easier to solve systems of linear equations. A matrix is in RREF if it satisfies the following conditions:

• All rows consisting entirely of zeros are grouped at the bottom of the matrix.
• Each row that is not entirely zeros has a leading entry (also called a pivot) that is to the right of the leading entry of the row above it.
• The leading entry in each row is 1.
• All other entries in the column of a leading entry are zeros.

Step 1: Write the Augmented Matrix

The first step in achieving RREF is to write the augmented matrix. An augmented matrix is a matrix that combines the coefficients of the variables in the system of linear equations with the constants on the right-hand side of the equations. To write the augmented matrix, start by writing the coefficients of the variables in a matrix, and then add an extra column to the right for the constants.

Example

Suppose we have the following system of linear equations:

2x + 3y - z = 4 x - 2y + 4z = -3 3x + 2y + z = 5

The augmented matrix for this system is:

[ 2  3 -1 | 4 ]
[ 1 -2  4 | -3 ]
[ 3  2  1 | 5 ]

Step 2: Perform Row Operations

Once you have the augmented matrix, the next step is to perform row operations to get the matrix into RREF. There are three types of row operations:

• Swap two rows.
• Multiply a row by a non-zero constant.
• Add a multiple of one row to another row.

These row operations can be used to transform the matrix into RREF.

Example

Using the augmented matrix from the previous example, we can perform the following row operations to get the matrix into RREF:

• Swap rows 1 and 2.
• Multiply row 2 by -2.
• Add 3 times row 2 to row 1.
• Add -4 times row 2 to row 3.

The resulting matrix is:

[ 1  0  0 | 1 ]
[ 0  1  0 | -2 ]
[ 0  0  1 | 3 ]

Step 3: Identify Leading Entries

Once you have performed the row operations, the next step is to identify the leading entries in each row. A leading entry is the first non-zero entry in a row. In RREF, the leading entry in each row is 1, and all other entries in the column of the leading entry are zeros.

Example

Using the matrix from the previous example, we can identify the leading entries as follows:

• Row 1: Leading entry is 1 in column 1.
• Row 2: Leading entry is 1 in column 2.
• Row 3: Leading entry is 1 in column 3.

Step 4: Eliminate Above Leading Entries

Once you have identified the leading entries, the next step is to eliminate any non-zero entries above the leading entries. This can be done by adding multiples of the row containing the leading entry to the rows above it.

Example

Using the matrix from the previous example, we can eliminate the non-zero entries above the leading entries as follows:

• Add -2 times row 1 to row 2.
• Add -3 times row 1 to row 3.

The resulting matrix is:

[ 1  0  0 | 1 ]
[ 0  1  0 | -2 ]
[ 0  0  1 | 3 ]

Step 5: Check for Reduced Row Echelon Form

The final step is to check if the matrix is in RREF. A matrix is in RREF if it satisfies the following conditions:

• All rows consisting entirely of zeros are grouped at the bottom of the matrix.
• Each row that is not entirely zeros has a leading entry that is to the right of the leading entry of the row above it.
• The leading entry in each row is 1.
• All other entries in the column of a leading entry are zeros.

If the matrix satisfies these conditions, then it is in RREF.

Example

Using the matrix from the previous example, we can see that it satisfies all the conditions for RREF. Therefore, the matrix is in RREF.

We hope this article has helped you understand the 5 steps to achieve Reduced Row Echelon Form. Remember to practice, practice, practice to become proficient in using this powerful tool in linear algebra. If you have any questions or need further clarification, please don't hesitate to ask. Share this article with your friends and classmates who may also benefit from it. Happy learning!