Augmented Matrix Calculator

Augmented Matrix Solver (3×3 System)

Enter the coefficients for your system of linear equations in the form:
ax + by + cz = d

Solution:

function solveMatrix() { var matrix = [ [parseFloat(document.getElementById('m00').value), parseFloat(document.getElementById('m01').value), parseFloat(document.getElementById('m02').value), parseFloat(document.getElementById('m03').value)], [parseFloat(document.getElementById('m10').value), parseFloat(document.getElementById('m11').value), parseFloat(document.getElementById('m12').value), parseFloat(document.getElementById('m13').value)], [parseFloat(document.getElementById('m20').value), parseFloat(document.getElementById('m21').value), parseFloat(document.getElementById('m22').value), parseFloat(document.getElementById('m23').value)] ]; var resultDiv = document.getElementById('matrix-result'); var output = document.getElementById('solution-output'); // Check for valid numbers for (var i = 0; i < 3; i++) { for (var j = 0; j < 4; j++) { if (isNaN(matrix[i][j])) { resultDiv.style.display = 'block'; output.innerHTML = 'Please fill in all fields with valid numbers.'; return; } } } var n = 3; // Gaussian Elimination logic for (var i = 0; i < n; i++) { // Search for maximum in this column var maxEl = Math.abs(matrix[i][i]); var maxRow = i; for (var k = i + 1; k maxEl) { maxEl = Math.abs(matrix[k][i]); maxRow = k; } } // Swap maximum row with current row (pivoting) for (var k = i; k < n + 1; k++) { var tmp = matrix[maxRow][k]; matrix[maxRow][k] = matrix[i][k]; matrix[i][k] = tmp; } // Check for singular matrix if (Math.abs(matrix[i][i]) < 1e-10) { resultDiv.style.display = 'block'; output.innerHTML = 'The system has no unique solution (Singular Matrix).'; return; } // Make all rows below this one 0 in current column for (var k = i + 1; k < n; k++) { var c = -matrix[k][i] / matrix[i][i]; for (var j = i; j = 0; i–) { res[i] = matrix[i][n] / matrix[i][i]; for (var k = i – 1; k >= 0; k–) { matrix[k][n] -= matrix[k][i] * res[i]; } } resultDiv.style.display = 'block'; output.innerHTML = 'x = ' + res[0].toFixed(4) + " + 'y = ' + res[1].toFixed(4) + " + 'z = ' + res[2].toFixed(4); }

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Understanding the Augmented Matrix

In linear algebra, an augmented matrix is a concise way of representing a system of linear equations. It essentially combines the coefficient matrix and the constant vector into a single grid, separated visually or conceptually by a vertical line. This representation is the primary starting point for using Gaussian Elimination to solve for unknown variables.

How an Augmented Matrix Works

Consider the following system of three equations with three variables:

• 2x + 1y – 1z = 8
• -3x – 1y + 2z = -11
• -2x + 1y + 2z = -3

To convert this into an augmented matrix, we strip away the variables (x, y, z) and the equals signs, leaving only the coefficients and the constants on the right side:

[ 2  1 -1 |  8 ]
[-3 -1  2 | -11 ]
[-2  1  2 | -3 ]

Solving Steps Using Row Operations

The goal is to transform the left side of the matrix into the Identity Matrix (ones on the diagonal and zeros elsewhere) using three types of elementary row operations:

1. Swapping: Swapping two rows.
2. Scaling: Multiplying a row by a non-zero constant.
3. Pivoting (Row Addition): Adding a multiple of one row to another row.

Example Calculation

Let's solve the simple system:
x + y = 5
2x – y = 1

1. Write the Matrix:
[ 1 1 | 5 ]
[ 2 -1 | 1 ]

2. Eliminate x from Row 2: Multiply Row 1 by -2 and add to Row 2.
[ 1 1 | 5 ]
[ 0 -3 | -9 ]

3. Solve for y: Divide Row 2 by -3.
[ 1 1 | 5 ]
[ 0 1 | 3 ] (So, y = 3)

4. Back Substitute: From Row 1: x + (3) = 5. Therefore, x = 2.

When is a Matrix "Inconsistent"?

If you perform row operations and end up with a row that looks like [0 0 0 | 5], it implies that 0 = 5, which is impossible. This means the system of equations has no solution. If you get a row like [0 0 0 | 0], it may indicate infinitely many solutions.