Reduced Row Echelon Form Calculator Symbolab

Reduced Row Echelon Form (RREF):

\[ \text{A matrix is in RREF when:} \] \[ 1. \text{Leading 1 in each row} \] \[ 2. \text{Zeros above and below each leading 1} \] \[ 3. \text{Leading 1's shift right in successive rows} \]

Reduced Row Echelon Form:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Reduced Row Echelon Form?

Reduced Row Echelon Form (RREF) is a special form of a matrix obtained by performing Gauss-Jordan elimination. It has leading 1's with zeros both above and below each leading 1, making it useful for solving systems of linear equations.

2. How Does the Calculator Work?

The calculator performs Gauss-Jordan elimination:

\[ \text{Steps:} \] \[ 1. \text{Forward elimination to echelon form} \] \[ 2. \text{Back substitution to reduced form} \] \[ 3. \text{Scaling to get leading 1's} \]

Key Features:

Handles any size matrix
Performs exact arithmetic (when possible)
Shows step-by-step solution (in advanced versions)

3. Importance of RREF

Applications: Solving linear systems, finding matrix rank, determining linear independence, computing inverses, and more in linear algebra.

4. Using the Calculator

Instructions: Enter your matrix with rows separated by newlines and elements separated by spaces or commas. For example:

1 2 3
4 5 6
7 8 9

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between REF and RREF?
A: REF has zeros below leading coefficients only, while RREF also has zeros above them and scaled leading 1's.

Q2: Can every matrix be reduced to RREF?
A: Yes, every matrix has a unique RREF, though the row operations to get there may vary.

Q3: How does RREF help solve linear systems?
A: The RREF directly shows solutions (or lack thereof) to the corresponding system of equations.

Q4: What does it mean if RREF has a row of zeros?
A: It indicates linear dependence among the original equations/vectors.

Q5: Can RREF be used to find matrix inverse?
A: Yes, by augmenting with the identity matrix and reducing to [I|A⁻¹].