Solving Matrix Calculator

2×2 System of Linear Equations Solver

Use this calculator to solve a system of two linear equations with two variables (x and y) using matrix methods (Cramer's Rule).

Enter the coefficients for your equations in the form:

a1*x + b1*y = c1

a2*x + b2*y = c2

Equation 1:

Equation 2:

function calculateMatrixSolution() { var a1 = parseFloat(document.getElementById('a1').value); var b1 = parseFloat(document.getElementById('b1').value); var c1 = parseFloat(document.getElementById('c1').value); var a2 = parseFloat(document.getElementById('a2').value); var b2 = parseFloat(document.getElementById('b2').value); var c2 = parseFloat(document.getElementById('c2').value); var resultDiv = document.getElementById('matrixResult'); // Input validation if (isNaN(a1) || isNaN(b1) || isNaN(c1) || isNaN(a2) || isNaN(b2) || isNaN(c2)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients."; return; } // Calculate the determinant of the coefficient matrix A // A = [[a1, b1], [a2, b2]] var detA = (a1 * b2) – (b1 * a2); var output = "

Input Equations:

"; output += "" + a1 + "x + " + b1 + "y = " + c1 + ""; output += "" + a2 + "x + " + b2 + "y = " + c2 + ""; output += "

Calculations:

"; output += "Determinant of Coefficient Matrix (detA) = (" + a1 + " * " + b2 + ") – (" + b1 + " * " + a2 + ") = " + detA.toFixed(4) + ""; if (detA === 0) { // Calculate determinants for Ax and Ay to check for infinite/no solutions var detAx = (c1 * b2) – (b1 * c2); var detAy = (a1 * c2) – (c1 * a2); if (detAx === 0 && detAy === 0) { output += "Since detA = 0, detAx = 0, and detAy = 0, there are infinitely many solutions (the lines are identical)."; } else { output += "Since detA = 0 but detAx or detAy is not zero, there is no unique solution (the lines are parallel and distinct)."; } } else { // Calculate determinant of Ax (replace x-column with constants) // Ax = [[c1, b1], [c2, b2]] var detAx = (c1 * b2) – (b1 * c2); output += "Determinant of Ax (detAx) = (" + c1 + " * " + b2 + ") – (" + b1 + " * " + c2 + ") = " + detAx.toFixed(4) + ""; // Calculate determinant of Ay (replace y-column with constants) // Ay = [[a1, c1], [a2, c2]] var detAy = (a1 * c2) – (c1 * a2); output += "Determinant of Ay (detAy) = (" + a1 + " * " + c2 + ") – (" + c1 + " * " + a2 + ") = " + detAy.toFixed(4) + ""; // Calculate x and y using Cramer's Rule var x = detAx / detA; var y = detAy / detA; output += "

Solution:

"; output += "x = detAx / detA = " + detAx.toFixed(4) + " / " + detA.toFixed(4) + " = " + x.toFixed(4) + ""; output += "y = detAy / detA = " + detAy.toFixed(4) + " / " + detA.toFixed(4) + " = " + y.toFixed(4) + ""; } resultDiv.innerHTML = output; } .matrix-solver-calculator { font-family: 'Arial', sans-serif; background-color: #f9f9f9; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; box-shadow: 0 2px 5px rgba(0,0,0,0.1); } .matrix-solver-calculator h2, .matrix-solver-calculator h3 { color: #333; text-align: center; margin-bottom: 15px; } .matrix-solver-calculator p { color: #555; line-height: 1.6; margin-bottom: 10px; } .calculator-inputs label { display: inline-block; width: 180px; margin-bottom: 8px; color: #444; } .calculator-inputs input[type="number"] { width: calc(100% – 200px); padding: 8px; margin-bottom: 10px; border: 1px solid #ddd; border-radius: 4px; box-sizing: border-box; } .calculator-inputs button { display: block; width: 100%; padding: 10px 15px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 16px; cursor: pointer; margin-top: 20px; transition: background-color 0.3s ease; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-results { margin-top: 25px; padding-top: 15px; border-top: 1px solid #eee; } .calculator-results h4 { color: #333; margin-bottom: 10px; } .calculator-results p { background-color: #e9ecef; padding: 10px; border-radius: 4px; border: 1px solid #dee2e6; margin-bottom: 8px; word-wrap: break-word; } .calculator-results p strong { color: #007bff; }

Understanding Systems of Linear Equations and Matrix Solutions

A system of linear equations is a collection of two or more linear equations involving the same set of variables. For example, a 2×2 system involves two equations and two variables, typically 'x' and 'y'. These systems are fundamental in mathematics, science, engineering, and economics, representing relationships between quantities.

What is a System of Linear Equations?

Consider two linear equations:

a1*x + b1*y = c1

a2*x + b2*y = c2

Here, a1, b1, c1, a2, b2, c2 are known coefficients and constants, while x and y are the variables we aim to solve for. A solution to this system is a pair of values (x, y) that satisfies both equations simultaneously.

Why Use Matrices to Solve Them?

Matrices provide a powerful and organized way to represent and solve systems of linear equations, especially as the number of equations and variables increases. For a 2×2 system, we can represent it in matrix form as AX = B, where:

• A is the coefficient matrix: [[a1, b1], [a2, b2]]
• X is the variable matrix: [[x], [y]]
• B is the constant matrix: [[c1], [c2]]

One common method for solving such systems using matrices is Cramer's Rule.

Cramer's Rule for 2×2 Systems

Cramer's Rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, provided that the determinant of the system's matrix is non-zero. For our 2×2 system:

1. Calculate the Determinant of the Coefficient Matrix (detA):
   detA = (a1 * b2) - (b1 * a2)
   This determinant is crucial. If detA = 0, there is no unique solution.
2. Calculate the Determinant of Ax (detAx):
   Replace the first column of matrix A (the 'x' coefficients) with the constant terms from matrix B.
   Ax = [[c1, b1], [c2, b2]]
detAx = (c1 * b2) - (b1 * c2)
3. Calculate the Determinant of Ay (detAy):
   Replace the second column of matrix A (the 'y' coefficients) with the constant terms from matrix B.
   Ay = [[a1, c1], [a2, c2]]
detAy = (a1 * c2) - (c1 * a2)
4. Find x and y:
   x = detAx / detA
y = detAy / detA

Interpreting the Determinant (detA)

• If detA ≠ 0: There is a unique solution (x, y). The two lines intersect at a single point.
• If detA = 0:
  • If detAx = 0 AND detAy = 0: There are infinitely many solutions. The two equations represent the same line.
  • If detAx ≠ 0 OR detAy ≠ 0: There is no solution. The two lines are parallel and distinct, meaning they never intersect.

Example Calculation:

Let's solve the system:

2x + 3y = 7

4x - 2y = 2

Here, a1=2, b1=3, c1=7 and a2=4, b2=-2, c2=2.

1. detA: (2 * -2) - (3 * 4) = -4 - 12 = -16
2. detAx: (7 * -2) - (3 * 2) = -14 - 6 = -20
3. detAy: (2 * 2) - (7 * 4) = 4 - 28 = -24
4. Solution:
   x = detAx / detA = -20 / -16 = 1.25
y = detAy / detA = -24 / -16 = 1.5

Thus, the unique solution to this system is x = 1.25 and y = 1.5. You can verify this by plugging these values back into the original equations.