f(x) GraphLab 3D 2D/3D Graphing Studio, Step-by-Step Matrix & Symbolic Calculus Workbench العربية

Exact Rational Step-by-Step Matrix & Linear Algebra Solver

Enter integer, decimal, or fraction cells (e.g. 1/3) and view every elementary row operation in simplified rational arithmetic p/q.

Exact Algebraic Solution & Step-by-Step Row Reduction Ledger
18 Worked Linear Algebra & Matrix Benchmark Solvers
Row Reduction & Echelon Form
Gauss–Jordan RREF Calculator (Step-by-Step Exact Fraction Row Operations)

Transforms any matrix into canonical Reduced Row Echelon Form using exact GCD-simplified rational elementary row operations.

3×4 · RREF
Linear Systems Ax = b
3×3 Linear System Solver Ax = b — Exact Augmented Matrix Elimination

Solves 2x₁ + x₂ − x₃ = 8, −3x₁ − x₂ + 2x₃ = −11, −2x₁ + x₂ + 2x₃ = −3 to exact solution (2, 3, −1) with zero residual.

3×3 · SYSTEM
Engineering & Mesh Analysis
4×4 Symmetric Laplacian Linear System — Heat Grid & Kirchhoff Node Solver

Solves diagonally dominant 4-node steady-state Laplace finite-difference / resistor mesh equations with exact rational fractions.

4×4 · SYSTEM
Parametric Solution Spaces
Infinite-Solutions Parametric System Solver — Particular + Free-Variable Basis

Identifies pivot vs. free variables when rank(A) = rank([A|b]) < n and constructs full vector solution x = x_p + t₁v₁ + t₂v₂.

3×4 · SYSTEM
Rank & Consistency Diagnosis
Inconsistent Linear System Detector — Rouché–Capelli Contradiction Proof

Demonstrates step-by-step how rank(A) < rank([A|b]) produces an explicit contradiction row 0 = 4, proving zero solutions exist.

3×3 · SYSTEM
Determinants & Volume Scaling
3×3 Determinant Calculator — Upper-Triangular Pivot Reduction & Sign Tracking

Computes exact determinant via Gaussian elimination while tracking row-swap parity (−1)^s and parallelepiped volume orientation.

3×3 · DET
Higher-Order Determinants
4×4 Matrix Determinant Calculator — O(n³) Pivot Elimination vs. 24 Permutations

Avoids 24-term Leibniz cofactor expansion by reducing 4×4 matrices cleanly to four exact rational diagonal pivots.

4×4 · DET
Matrix Inversion
3×3 Matrix Inverse Calculator [A | I] → [I | A⁻¹] — Exact Rational Entries

Augments A with the identity matrix I₃ and executes full Gauss–Jordan elimination to extract exact rational A⁻¹.

3×3 · INVERSE
Numerical Conditioning Benchmarks
Hilbert 3×3 Matrix H₃ Exact Rational Inverse — Why Exact Fractions Beat Floats

Recovers exact integer inverse [[9,−36,30],[−36,192,−180],[30,−180,180]] and det(H₃) = 1/2160 without floating-point rounding loss.

3×3 · INVERSE
Rank Deficiency & Singularity
Singular Matrix Rank Diagnosis — Why [[1,2,3],[4,5,6],[7,8,9]] Has No Inverse

Reveals R₃ − 2R₂ + R₁ = 0 during row reduction, proving rank(A) = 2 < 3 and det(A) = 0.

3×3 · INVERSE
Spectral Theory & Eigenvalues
3×3 Symmetric Matrix Eigenvalues, Characteristic Polynomial & Eigenvectors

Computes characteristic polynomial λ³ − 12λ² + 45λ − 54 = 0, spectral eigenvalues λ = 6, 3, 3, and real eigenvectors.

3×3 · EIGEN
Complex Eigenvalues & Dynamical Systems
2×2 Rotation–Scaling Matrix Complex Conjugate Eigenvalues λ = 3 ± 4i

Explains why spiral rotation-dilation matrices have negative discriminant Δ < 0 and complex conjugate eigenvalues α ± βi.

2×2 · EIGEN
Stochastic Processes & Markov Chains
Markov Transition Matrix Perron–Frobenius Eigenvalue λ₁ = 1 & Mixing Rate

Analyzes doubly stochastic transition matrices where dominant eigenvalue λ₁ = 1 governs stationary equilibrium and |λ₂| controls convergence speed.

3×3 · EIGEN
Fundamental Subspaces & Rank-Nullity
Nullspace N(A) & Column Space C(A) Basis Calculator — Rank-Nullity Theorem

Extracts pivot columns of A for Col(A) and special free-variable vectors for Null(A), verifying rank(A) + nullity(A) = n.

3×4 · SUBSPACES
Matrix Factorizations
PA = LU Matrix Factorization Calculator with Partial Row Pivoting

Decomposes square matrices into permutation P, unit lower-triangular L, and upper-triangular U factors for fast multi-RHS solves.

3×3 · LU
Orthogonalization & Least Squares
Modified Gram–Schmidt QR Orthogonalization Calculator (A = Q·R)

Orthonormalizes the columns of classic Golub–Van Loan benchmark [[12,−51,4],[6,167,−68],[−4,24,−41]] into orthogonal Q and upper-triangular R.

3×3 · QR
Polynomial Interpolation
4×4 Vandermonde Interpolation System — Exact Cubic Curve Fitting

Determines exact cubic coefficients c₀ + c₁x + c₂x² + c₃x³ passing through (−1,2), (0,1), (1,2), and (2,11).

4×4 · SYSTEM
Combinatorial Matrices
4×4 Symmetric Pascal Matrix — Unimodular det(P₄) = 1 & Triangular Factors

Binomial-coefficient matrix Pᵢⱼ = C(i+j, i) whose LU triangular factors have unit diagonals, forcing det(Pₙ) = 1 for every dimension n.

4×4 · LU

Mathematical Reference & Frequently Asked Questions

Why does exact rational p/q arithmetic matter in Gauss–Jordan elimination?

Floating-point rounding corrupts pivot tests on ill-conditioned matrices like the Hilbert matrix H₃. Exact GCD-simplified BigInt rationals preserve exact zeros and exact integer inverses.

How does the solver classify unique, infinite, and inconsistent Ax = b systems?

Using Rouché–Capelli rank analysis on [A | b]: if rank(A) < rank([A | b]) it isolates the contradiction row 0 = c; if rank(A) = rank([A | b]) < n it outputs the particular vector plus free-variable nullspace basis vectors.