RREF Calculator

The RREF calculator simplifies and organizes a system of linear equations represented in matrix form and transforms them into a reduced row echelon form. This is done by applying a series of row operations such as swapping rows, multiplying rows by non-zero constants, and adding multiples of one row to another.

How to use this reduced row echelon form calculator?

Calculating the Reduced Row Echelon Form (RREF) using our RREF calculator is quite simple, It only requires a few inputs to perform, including:

• First, Set the size of the matrix for which you are going to calculate

• Then, Enter the values of the matrix in the required field

• Finally, tap on the Calculate button

The RREF calculator will quickly process the information and provide you with the reduced echelon form of the matrix along with step-by-step solutions.

What is reduced row echelon form (RREF)?

The reduced row echelon form (RREF) is a standardized and simplified representation of a matrix achieved through a series of row operations being applied.

A matrix is in reduced row echelon form when it meets three conditions: 

• It's already in a row echelon form
• All its pivots (the first non-zero entry in each row) are 1
• The pivots are the only non-zero numbers in their respective columns

How do I find the RREF of a matrix?

Let's go through an example of finding the RREF of a matrix for better understanding, Here are the steps: 

Example:

Consider the following matrix:

\[ A = \begin{bmatrix} 2 & 1 & 3 \\ 1 & 2 & 1 \\ 3 & 3 & 5 \end{bmatrix} \]

Step 1: Divide row 1 by 2:

• \[ \begin{bmatrix}1 & 0.5 & 1.5 \\1 & 2 & 1 \\3 & 3 & 5\end{bmatrix} \]

Step 2: Subtract row 1 multiplied by 1 from row R2:

• \[ \begin{bmatrix}1 & 0.5 & 1.5 \\0 & 1.5 & -0.5 \\3 & 3 & 5\end{bmatrix} \]

Step 3: Subtract row 2 multiplied by 3 from row R3:

• \[ \begin{bmatrix}1 & 0.5 & 1.5 \\0 & 1.5 & -0.5 \\0 & -0.5 & 0.5\end{bmatrix} \]

Step 4: Multiply row 2 by 2/3:

• \[ \begin{bmatrix}1 & 0.5 & 1.5 \\0 & 1 & -1/3 \\0 & -0.5 & 0.5\end{bmatrix} \]

Step 5: Subtract row 0 multiplied by 1/2 from row R1:

• \[ \begin{bmatrix}1 & 0 & 0.5 \\0 & 1 & -1/3 \\0 & -0.5 & 0.5\end{bmatrix} \]

Step 6: Subtract row 2 multiplied by 1/2 from row R3:

• \[ \begin{bmatrix}1 & 0 & 0.5 \\0 & 1 & -1/3 \\0 & 0 & 1\end{bmatrix} \]

Step 7: Multiply row 3 by 3/2:

• \[ \begin{bmatrix}1 & 0 & 0.5 \\0 & 1 & -1/3 \\0 & 0 & 1\end{bmatrix} \]

Step 8: Subtract row 0 multiplied by 5/3 from row R1:

• \[ \begin{bmatrix}1 & 0 & 0 \\0 & 1 & -1/3 \\0 & 0 & 1\end{bmatrix} \]

Step 9: Subtract row 1 multiplied by -1/3 from row R2:

• \[ \begin{bmatrix}1 & 0 & 0 \\0 & 1 & 0 \\0 & 0 & 1\end{bmatrix} \]

So, this is the final reduced row echelon form of the given matrix. Now that you have gone through the process, we hope you have gained a clear understanding of how to determine the reduced row echelon form (RREF) of any matrix using the RREF calculator provided by Calculatored.