Use this Matrix Calculator to compute determinants, find inverses with step-by-step Gauss–Jordan operations, and multiply matrices (supports up to 6×6).
Matrix Calculator — Advanced
Determinant, Inverse (with Gauss–Jordan steps), and Multiplication. Supports square matrices up to 6×6. Chart visualization and MathJax explanations included.
Quick summary: A browser-based Matrix Calculator that computes determinants, finds inverses (with step-by-step Gauss–Jordan operations), and multiplies matrices. It supports square matrices up to 6×6 and rectangular matrices for multiplication. Includes visual table output, charts, and MathJax-rendered formulas.
What is this calculator?
This Matrix Calculator is an interactive tool that performs common matrix operations: calculating the determinant of a square matrix, computing the inverse of an invertible matrix using Gauss–Jordan elimination (with a human-readable step log), and multiplying two compatible matrices (A × B). It accepts integer and decimal inputs and displays results as styled HTML tables with optional chart visualization and formula explanations rendered via MathJax.
Why you might need it
Matrices are used everywhere in engineering, physics, data science, computer graphics, systems of linear equations, and machine learning. This calculator helps students, engineers, and hobbyists quickly validate calculations, learn row operation steps for the inverse, verify matrix products, and export results for reports or assignments.
Quick facts
– Supports square matrices from 2×2 up to 6×6 for determinant & inverse.
– Multiplication supports rectangular matrices (A rows/cols and B rows/cols up to a sensible cap).
– Inverse is computed via Gauss–Jordan elimination with partial pivoting and step-by-step logging for teaching.
– Determinant uses recursive expansion by minors (accurate but slower for 5×5 / 6×6 — LU decomposition is available on request for faster, more stable results).
– Outputs are downloadable as CSV and copyable to clipboard.
Real-life examples
Example 1 — Solving a system of linear equations: Convert a linear system Ax = b into matrix form and compute A⁻¹ (if invertible) to find x = A⁻¹b.
Example 2 — Computer graphics: Use transformation matrices (rotation, scaling, translation) and multiply them to apply combined transforms to 2D/3D points.
Example 3 — Engineering: Compute determinants to check whether a set of vectors is linearly independent (non-zero determinant means independence for square matrices).
How to use the calculator
1. Choose the operation: Determinant, Inverse, or Multiply (A × B).
2. For Determinant/Inverse — select the matrix size (2–6) and enter values into the matrix grid.
3. For Multiply — set dimensions for A and B (A columns must equal B rows), enter values for both matrices.
4. Click Calculate. View the table output, steps (for inverse), charts, and use Copy/Export if needed.
5. Use Reset to restore defaults.
Tip: Press Enter inside a cell to trigger calculation (works in most browsers).
Formulas & methods (summary)
Determinant (2×2):
For matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$,
$$\det(A) = ad – bc.$$
Determinant (3×3) — Sarrus’ rule (visual):
For $A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$,
$$\det(A) = aei + bfg + cdh – ceg – bdi – afh.$$
Determinant (n×n):
Computed by expansion by minors (Laplace expansion) or via LU decomposition (optional). The calculator currently uses recursive expansion by minors (good for teaching, slower for n ≥ 5).
Inverse — Gauss–Jordan method:
To compute $A^{-1}$, form the augmented matrix $[A|I]$, apply row swaps, scaling, and eliminations until the left side becomes $I$, producing $[I|A^{-1}]$.
Why the Gauss–Jordan method?
Gauss–Jordan gives both the inverse and a step-by-step sequence of row operations, which is excellent for learning and verification. For large matrices or performance-sensitive tasks, LU decomposition is faster and more numerically stable — tell me if you want that implemented.
Tips & best practices
– For symbolic or exact fraction results, consider entering rational values and requesting fraction output (feature available on request).
– For 5×5 and 6×6 determinants, expect a longer compute time — use LU decomposition if you need speed.
– If determinant ≈ 0 (very small absolute value), numerical round-off could make the matrix effectively singular — check inputs and scale values if necessary.
– Use the CSV export to download results for further analysis in Excel or Google Sheets.
Frequently Asked Questions (FAQ)
Q: What if the calculator says the matrix is singular?
A: “Singular” means the matrix does not have an inverse — its determinant is zero (or extremely close to zero numerically). This usually means the rows/columns are linearly dependent. Double-check your inputs.
Q: Why does the determinant for a 6×6 take a while?
A: The current determinant method is recursive expansion by minors. Complexity grows factorially; computing a 6×6 determinant by recursion is CPU-heavy in JavaScript. If you need faster results, request LU decomposition — I can update the calculator to use that.
Q: Can this tool show how to solve systems of equations step-by-step?
A: Yes — using the Gauss–Jordan steps for the inverse helps; additionally I can add a mode where you input matrix A and vector b, and the calculator will show explicit row operations solving [A|b] directly.
Export, copy & charts
The calculator supports:
- Copy result to clipboard
- Export matrix and results to CSV
- Chart visualization (flattened cell values as a bar chart). A heatmap-style display can be added on request.
History & background (short)
Matrices were formalized in the 19th century as mathematicians developed linear algebra — with key contributions from Cayley and Sylvester. Determinants date earlier and were studied in solving systems of linear equations and for properties of linear transformations. Gauss and Jordan developed elimination ideas in the 18th–19th centuries, now core to modern linear algebra algorithms.
Disclaimer
For educational and convenience purposes only. While the calculator is designed to provide correct numeric results, rounding and floating-point arithmetic can cause minor numerical errors, especially for near-singular matrices or very large/small values. For critical numerical work (scientific computing, production systems), validate using a trusted numerical library (e.g., LAPACK, NumPy) and consider using LU or QR decompositions for better stability. Use outputs at your own risk.