Reduced Row Echelon Form (RREF) Calculator
Enter the coefficients of your matrix below. This calculator supports a 3×4 matrix, suitable for solving systems of 3 linear equations with 3 variables (e.g., Ax + By + Cz = D).
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Resulting Reduced Row Echelon Form:
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for (j = 0; j < cols; j++) {
// Round to a reasonable number of decimal places to avoid floating point artifacts
var displayVal = Math.round(matrix[i][j] * 1e6) / 1e6;
outputHtml += "
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}
resultDiv.innerHTML = outputHtml;
}
// Run calculation on page load for initial values
window.onload = calculateRREF;
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if (j === cols – 2) { // After the third column (index 2) for a 3×4 matrix
outputHtml += "|";
}
}
outputHtml += "Understanding Reduced Row Echelon Form (RREF)
The Reduced Row Echelon Form (RREF) is a standardized way to represent a matrix, particularly useful in linear algebra for solving systems of linear equations, finding the inverse of a matrix, and determining the rank of a matrix. It's the final output of the Gauss-Jordan elimination process.
Key Properties of RREF:
- All-zero rows are at the bottom: If a row consists entirely of zeros, it is at the bottom of the matrix.
- Leading 1s: The first non-zero entry in each non-zero row (called the "leading entry" or "pivot") is a 1.
- Staircase pattern: Each leading 1 is to the right of the leading 1 in the row above it.
- Zeroes above and below leading 1s: Each column that contains a leading 1 has zeros everywhere else in that column.
Why is RREF Important?
- Solving Systems of Linear Equations: When an augmented matrix (a matrix representing a system of equations) is transformed into RREF, the solution to the system can be read directly from the matrix. For example, if the RREF is:
1 0 0 | 5 0 1 0 | 3 0 0 1 | 1This directly translates to x=5, y=3, z=1. - Matrix Inversion: To find the inverse of a square matrix A, you can augment it with an identity matrix [A | I] and then apply Gauss-Jordan elimination to transform it into [I | A-1].
- Determining Matrix Rank: The rank of a matrix is the number of non-zero rows in its RREF.
- Basis for Vector Spaces: RREF helps in finding a basis for the row space, column space, and null space of a matrix.
How the Calculator Works (Briefly):
This calculator uses the Gauss-Jordan elimination method to transform your input matrix into its Reduced Row Echelon Form. The process involves a series of elementary row operations:
- Swapping two rows: To get a non-zero pivot element.
- Multiplying a row by a non-zero scalar: To make a leading entry equal to 1.
- Adding a multiple of one row to another row: To create zeros above and below the leading 1s.
These operations are applied systematically to achieve the RREF properties. The calculator handles a 3×4 matrix, which is common for systems of three linear equations with three variables.
Example Usage:
Consider the following system of linear equations:
x + 2y - z = -4
2x + 3y - z = -11
-2x + 0y - 3z = 22
This system can be represented by the augmented matrix:
[ 1 2 -1 | -4 ]
[ 2 3 -1 | -11 ]
[ -2 0 -3 | 22 ]
Enter these values into the calculator's input fields. After clicking "Calculate RREF", the output will be:
[ 1 0 0 | -1 ]
[ 0 1 0 | -2 ]
[ 0 0 1 | -6 ]
From this RREF, we can directly read the solution: x = -1, y = -2, and z = -6.