Reduced Row Echelon Form Calculator

Reduced Row Echelon Form Calculator

Enter the elements of your 3×4 augmented matrix below. This calculator will transform it into its Reduced Row Echelon Form (RREF).

What is Reduced Row Echelon Form (RREF)?

Reduced Row Echelon Form (RREF) is a specific, unique form that any matrix can be transformed into using a series of elementary row operations. It is a fundamental concept in linear algebra, primarily used for solving systems of linear equations, determining the rank of a matrix, and finding the inverse of a square matrix.

Key Properties of RREF:

• Leading Entry (Pivot): In each non-zero row, the first non-zero entry (reading from left to right) is called the leading entry or pivot.
• Leading Entry is 1: Every leading entry must be 1.
• Zeroes Below and Above Pivots: Each column that contains a leading entry (a '1') must have zeros in all other positions within that column.
• Staircase Pattern: The leading entry of any row is always to the right of the leading entry of the row directly above it.
• Zero Rows at Bottom: All rows consisting entirely of zeros (if any) are located at the bottom of the matrix.

How is RREF Achieved? (Gauss-Jordan Elimination)

The process of transforming a matrix into its RREF is known as Gauss-Jordan elimination. This method systematically applies three types of elementary row operations:

1. Row Swapping: Interchanging two rows (e.g., R1 ↔ R2). This is used to bring a non-zero pivot to the correct position.
2. Row Scaling: Multiplying a row by a non-zero scalar (e.g., 2R1 → R1). This is used to make a pivot equal to 1.
3. Row Addition: Adding a multiple of one row to another row (e.g., R2 + 3R1 → R2). This is used to create zeros above and below pivots.

The algorithm typically proceeds in two main phases:

1. Forward Elimination (to Row Echelon Form):
   • Starting from the first column, identify the first non-zero entry. If it's not in the first row, swap rows to bring it there. This becomes your first pivot.
   • Scale this row so that the pivot becomes 1.
   • Use row addition operations to make all entries below this pivot zero.
   • Move to the next column to the right and repeat the process for the submatrix below the current row, working downwards until the matrix is in Row Echelon Form (all entries below pivots are zero).
2. Backward Elimination (to Reduced Row Echelon Form):
   • Starting from the last non-zero row and working upwards, use the leading 1s (pivots) to make all entries above them zero.
   • Continue this process until all entries above every pivot are zero, resulting in the Reduced Row Echelon Form.

Example of RREF Calculation:

Consider the following system of linear equations:

x + 2y -  z = -4
2x + 3y -  z = -11
-x + 0y - 2z =  7
        

This system can be represented by the following augmented matrix:

[ 1  2 -1 | -4 ]
[ 2  3 -1 | -11 ]
[ -1 0 -2 |  7 ]
        

Applying the Gauss-Jordan elimination steps (as performed by this calculator), the matrix is transformed into its Reduced Row Echelon Form:

[ 1  0  0 | -13 ]
[ 0  1  0 |   6 ]
[ 0  0  1 |   3 ]
        

From this RREF, we can directly read the unique solution to the system of equations: x = -13, y = 6, and z = 3.

Reduced Row Echelon Form:

' + val.toFixed(4) + '

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