Augmented Matrix Calculator: Solving Systems of Linear Equations
An augmented matrix is a powerful mathematical tool used to represent and solve systems of linear equations. It combines the coefficient matrix of a system of equations with the constant terms into a single matrix. This compact notation simplifies the process of applying various matrix operations, such as Gaussian elimination or Gauss-Jordan elimination, to find the values of the unknown variables.
What is an Augmented Matrix?
Consider a system of linear equations with three variables (x, y, z):
a₁₁x + a₁₂y + a₁₃z = d₁ a₂₁x + a₂₂y + a₂₃z = d₂ a₃₁x + a₃₂y + a₃₃z = d₃
This system can be represented as an augmented matrix in the following form:
[ a₁₁ a₁₂ a₁₃ | d₁ ] [ a₂₁ a₂₂ a₂₃ | d₂ ] [ a₃₁ a₃₂ a₃₃ | d₃ ]
The vertical line separates the coefficient matrix (the left part) from the column vector of constants (the right part).
How This Calculator Works (Cramer's Rule)
While Gaussian elimination is a general method, for square systems of linear equations with a unique solution, Cramer's Rule provides a direct way to find the values of the variables using determinants. This calculator utilizes Cramer's Rule for 3×3 systems (3 equations, 3 variables).
Cramer's Rule states that for a system:
a₁₁x + a₁₂y + a₁₃z = d₁ a₂₁x + a₂₂y + a₂₃z = d₂ a₃₁x + a₃₂y + a₃₃z = d₃
The solutions for x, y, and z are given by:
x = Dₓ / D y = Dᵧ / D z = D₂ / D
Where:
Dis the determinant of the coefficient matrix.Dₓis the determinant of the matrix formed by replacing the x-coefficient column (first column) in the coefficient matrix with the constant terms (d₁, d₂, d₃).Dᵧis the determinant of the matrix formed by replacing the y-coefficient column (second column) in the coefficient matrix with the constant terms (d₁, d₂, d₃).D₂is the determinant of the matrix formed by replacing the z-coefficient column (third column) in the coefficient matrix with the constant terms (d₁, d₂, d₃).
If D = 0, the system either has no unique solution (infinite solutions or no solution at all), and Cramer's Rule cannot be directly applied to find a unique solution.
Augmented Matrix Calculator
Enter the coefficients and constants for your 3×3 system of linear equations below. The calculator will use Cramer's Rule to find the values of x, y, and z.
Example Calculation
Let's use the default values provided in the calculator:
2x + 1y - 1z = 8 -3x - 1y + 2z = -11 -2x + 1y + 2z = -3
Inputting these values into the calculator will yield the solution for x, y, and z as x=2, y=3, z=-1.
function calculateAugmentedMatrix() { // Helper function to get a numeric value from an input field function getInputValue(id) { var value = parseFloat(document.getElementById(id).value); if (isNaN(value)) { return 0; // Default to 0 for missing/invalid inputs } return value; } // Get all matrix elements var a11 = getInputValue("a11"); var a12 = getInputValue("a12"); var a13 = getInputValue("a13"); var d1 = getInputValue("d1"); var a21 = getInputValue("a21"); var a22 = getInputValue("a22"); var a23 = getInputValue("a23"); var d2 = getInputValue("d2"); var a31 = getInputValue("a31"); var a32 = getInputValue("a32"); var a33 = getInputValue("a33"); var d3 = getInputValue("d3"); // Display the original system var originalSystem = ` Original System: ${a11}x + ${a12}y + ${a13}z = ${d1} ${a21}x + ${a22}y + ${a23}z = ${d2} ${a31}x + ${a32}y + ${a33}z = ${d3} `; // Function to calculate the determinant of a 3×3 matrix function calculateDeterminant(m11, m12, m13, m21, m22, m23, m31, m32, m33) { return m11 * (m22 * m33 – m23 * m32) – m12 * (m21 * m33 – m23 * m31) + m13 * (m21 * m32 – m22 * m31); } // Calculate D (determinant of the coefficient matrix) var D = calculateDeterminant(a11, a12, a13, a21, a22, a23, a31, a32, a33); var resultDiv = document.getElementById("result"); var output = originalSystem; if (D === 0) { output += "Determinant (D) = 0. This system either has no unique solution (infinite solutions or no solution). Cramer's Rule cannot be used to find a unique solution."; } else { // Calculate Dx var Dx = calculateDeterminant(d1, a12, a13, d2, a22, a23, d3, a32, a33); // Calculate Dy var Dy = calculateDeterminant(a11, d1, a13, a21, d2, a23, a31, d3, a33); // Calculate Dz var Dz = calculateDeterminant(a11, a12, d1, a21, a22, d2, a31, a32, d3); // Calculate x, y, z var x = Dx / D; var y = Dy / D; var z = Dz / D; output += ` Determinant (D): ${D.toFixed(4)} Determinant (Dx): ${Dx.toFixed(4)} Determinant (Dy): ${Dy.toFixed(4)} Determinant (Dz): ${Dz.toFixed(4)}