What is a linear system?

A set of equations in matrix form Ax=b where A is coefficient matrix, x is unknown vector, b is constants.

If A is invertible: x = A^-1 * b. Or use Cramer rule: x_i = det(A_i)/det(A).

When is there no solution?

When det(A)=0 (singular matrix). System has either no solutions or infinite solutions.

What is Cramers rule?

Replace column i of A with b, compute determinant, divide by det(A) to get x_i.

IP331.com | Online Tools

Algebra Tools Geometry Tools Trigonometry Tools Matrix Tools Function Tools Number Theory Tools Base Converter Electronics & Physics Statistics Tools Life & Finance

Home›Matrix Tools›Linear System Solver

Linear System Solver

Solve Ax = b for 2x2 and 3x3 systems of equations

Matrix A (coefficients)

Vector b (constants)

System Solving Method

Ax = b → x = A^(-1) * b (if A invertible)

Cramer rule: x_i = det(A_i) / det(A)

Check: A * x should equal b

No solution if det(A) = 0

A linear system has a unique solution when the coefficient matrix is invertible. If det=0, check consistency.

More Matrix Tools

Matrix Null Space Calculator

Find basis for null space (kernel) of Ax=0 via RREF

›

SVD Decomposition Calculator

Compute singular value decomposition A = UΣV* with detailed step-by-step computation

›

Left Null Space Calculator

Find left null space basis via transpose and RREF

›

Orthonormal Basis Generator

Generate orthonormal basis with Gram-Schmidt and normalization for 2D and 3D vectors

›

Idempotent Matrix Checker

Verify if A² = A (idempotent matrix) with detailed step-by-step verification

›

Matrix Characteristic Polynomial Calculator

Compute det(A-λI) for 2x2 and 3x3 matrices with detailed step-by-step derivation

›