1. What Is Reduced Row Echelon Form?

The Reduced Row Echelon Form (RREF) is a simplified version of a matrix obtained through Gauss-Jordan elimination. It has leading 1s (pivots) with zeros in all positions above and below each pivot.

2. How Does the Calculator Work?

The calculator performs Gauss-Jordan elimination to transform any matrix into its RREF:

Steps:

1. Identify the leftmost nonzero column (pivot column)
2. Select a nonzero entry in the pivot column as pivot
3. Swap rows to move pivot to current row
4. Make pivot equal to 1 by dividing the row
5. Create zeros above and below the pivot
6. Repeat for each pivot position

3. Importance of RREF

Applications: Solving systems of linear equations, determining matrix rank, finding matrix inverses, and analyzing vector spaces.

4. Using the Calculator

Instructions: Enter your matrix with each row on a new line and elements separated by spaces or commas. 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: RREF has zeros both below AND above each pivot, while REF only requires zeros below pivots.

Q2: Does every matrix have an RREF?
A: Yes, every matrix has exactly one unique RREF.

Q3: What does RREF reveal about a matrix?
A: It shows the rank (number of pivots), linear independence of rows/columns, and solutions to Ax=b.

Q4: Can RREF be used for any size matrix?
A: Yes, it works for any m×n matrix, including non-square matrices.

Q5: How is RREF different from matrix inversion?
A: RREF works for all matrices, while inversion is only possible for square, full-rank matrices.