5 Steps To Reduced Echelon Form

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Solving systems of linear equations is a fundamental concept in mathematics, and one of the most effective methods for doing so is by transforming the augmented matrix into Reduced Echelon Form (REF). This process involves a series of row operations that ultimately lead to a simplified matrix, making it easier to solve for the variables. In this article, we will delve into the 5 steps required to transform an augmented matrix into Reduced Echelon Form.

Understanding Reduced Echelon Form

Before we dive into the steps, it's essential to understand what Reduced Echelon Form is. A matrix is said to be in Reduced Echelon Form 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 1 as its first nonzero entry (this entry is called a leading entry or pivot).
• The column in which a leading 1 of a row is found has all zeros elsewhere, so a column containing a leading 1 will have zeros everywhere except for one place.

Benefits of Reduced Echelon Form

Reduced Echelon Form provides several benefits, including:

• Easy identification of the solution to the system of linear equations
• Simplified matrix operations
• Ability to determine the rank and nullity of the matrix

Step 1: Write the Augmented Matrix

The first step in transforming an augmented matrix into Reduced Echelon Form is to write the augmented matrix itself. The augmented matrix is a combination of the coefficient matrix and the constant matrix. For example, consider the system of linear equations:

2x + 3y - z = 5 x - 2y + 4z = -2 3x + y + 2z = 7

The augmented matrix for this system would be:

| 2 3 -1 | 5 | | 1 -2 4 | -2 | | 3 1 2 | 7 |

Row Operations

To transform the augmented matrix into Reduced Echelon Form, we need to perform a series of row operations. These operations include:

• Row switching: swapping two rows
• Row multiplication: multiplying a row by a non-zero constant
• Row addition: adding a multiple of one row to another row

Step 2: Get a Leading 1 in the Top Left Corner

The next step is to get a leading 1 in the top left corner of the matrix. To do this, we can use row operations to create a 1 in the top left corner. If the top left entry is not already a 1, we can multiply the row by a non-zero constant to make it a 1.

For example, consider the augmented matrix:

| 2 3 -1 | 5 | | 1 -2 4 | -2 | | 3 1 2 | 7 |

To get a leading 1 in the top left corner, we can multiply the first row by 1/2:

| 1 3/2 -1/2 | 5/2 | | 1 -2 4 | -2 | | 3 1 2 | 7 |

Step 3: Get Zeros Below the Leading 1

Once we have a leading 1 in the top left corner, the next step is to get zeros below the leading 1. To do this, we can use row operations to add multiples of the first row to the other rows.

For example, consider the augmented matrix:

| 1 3/2 -1/2 | 5/2 | | 1 -2 4 | -2 | | 3 1 2 | 7 |

To get zeros below the leading 1, we can add -1 times the first row to the second row, and -3 times the first row to the third row:

| 1 3/2 -1/2 | 5/2 | | 0 -7/2 9/2 | -9/2 | | 0 -7/2 7/2 | 5/2 |

Step 4: Get a Leading 1 in the Second Row

The next step is to get a leading 1 in the second row. To do this, we can use row operations to create a 1 in the second row. If the second row does not already have a leading 1, we can multiply the row by a non-zero constant to make it a 1.

For example, consider the augmented matrix:

| 1 3/2 -1/2 | 5/2 | | 0 -7/2 9/2 | -9/2 | | 0 -7/2 7/2 | 5/2 |

To get a leading 1 in the second row, we can multiply the second row by -2/7:

| 1 3/2 -1/2 | 5/2 | | 0 1 -9/7 | 9/7 | | 0 -7/2 7/2 | 5/2 |

Step 5: Get Zeros Below the Leading 1 in the Second Row

The final step is to get zeros below the leading 1 in the second row. To do this, we can use row operations to add multiples of the second row to the other rows.

For example, consider the augmented matrix:

| 1 3/2 -1/2 | 5/2 | | 0 1 -9/7 | 9/7 | | 0 -7/2 7/2 | 5/2 |

To get zeros below the leading 1 in the second row, we can add 7/2 times the second row to the third row:

| 1 3/2 -1/2 | 5/2 | | 0 1 -9/7 | 9/7 | | 0 0 1 | 2 |

By following these 5 steps, we can transform an augmented matrix into Reduced Echelon Form, making it easier to solve systems of linear equations.

We hope this article has been helpful in explaining the 5 steps to Reduced Echelon Form. If you have any questions or need further clarification, please don't hesitate to ask.