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Put Matrix in Reduced Row Echelon Form Calculator

Reviewed by Calculator Editorial Team

Reduced Row Echelon Form (RREF) is a standard form for matrices that simplifies solving systems of linear equations. This calculator will help you convert any matrix to its RREF form quickly and accurately.

What is Reduced Row Echelon Form?

A matrix is in Reduced Row Echelon Form (RREF) if it meets the following conditions:

  1. The first non-zero entry in each row (called the leading entry) is 1 (called a leading 1).
  2. Each leading 1 is to the right of the leading 1 in the row above it.
  3. All entries in a column containing a leading 1 are 0, except for the leading 1 itself.

RREF is particularly useful for solving systems of linear equations, finding the rank of a matrix, and determining the basis of a vector space.

How to Convert a Matrix to RREF

Converting a matrix to RREF involves a series of row operations:

  1. Row swapping: Swap any two rows.
  2. Row scaling: Multiply any row by a non-zero scalar.
  3. Row replacement: Add a multiple of one row to another row.

The process typically involves:

  1. Finding the leftmost non-zero column.
  2. Selecting a non-zero entry in this column as a pivot.
  3. Swapping rows to bring the pivot to the top.
  4. Scaling the pivot row to make the pivot element 1.
  5. Using row replacement to make all other entries in the pivot column 0.
  6. Repeating the process for the submatrix below the pivot.

Note: The RREF is unique for any given matrix, meaning there's only one correct RREF form for any matrix.

Worked Example

Let's convert the following matrix to RREF:

[ 1 2 3 ] [ 2 5 6 ] [ 3 7 9 ]

Step 1: The first pivot is already in the first row, first column.

Step 2: Subtract 2 times row 1 from row 2, and subtract 3 times row 1 from row 3:

[ 1 2 3 ] [ 0 1 0 ] [ 0 1 0 ]

Step 3: Subtract row 2 from row 3:

[ 1 2 3 ] [ 0 1 0 ] [ 0 0 0 ]

The final RREF form is:

[ 1 0 -1 ] [ 0 1 3 ] [ 0 0 0 ]

FAQ

What is the difference between row echelon form and reduced row echelon form?

Row Echelon Form (REF) requires the leading entries to be 1, but doesn't require the columns above leading 1s to be 0. RREF is stricter, requiring all entries above and below leading 1s to be 0.

Can any matrix be converted to RREF?

Yes, any matrix can be converted to RREF using elementary row operations. The process is guaranteed to terminate with a unique RREF form.

What are the applications of RREF?

RREF is used in solving systems of linear equations, finding the rank of a matrix, determining the basis of a vector space, and in various applications in linear algebra.