Matrix Multiplicative Inverse Calculator

Matrix Input & Options

Matrix size (n × n)

Decimal precision

Output format

Decimals

Fractions

Show elimination steps

Enter matrix A entries

Tips: use integers, decimals, or simple fractions like 3/7. Paste-ready and CSV upload helpers below.

Paste matrix (rows separated by new lines)

Upload CSV to populate matrix

CSV should contain an n × n grid. Extra cells are ignored.

Quick examples: Well-conditioned 3×3 Singular 3×3 (no inverse) Ill-conditioned 3×3

Quick Summary

Supports 2×2 up to 10×10 square matrices.
Accepts integers, decimals, and simple fractional inputs.
Uses Gauss–Jordan elimination with partial pivoting.
Computes determinant and infinity-norm condition number.
Verifies A × A⁻¹ ≈ I for numerical confidence.
Export inverse as CSV; print-friendly summary for PDF export.

For ill-conditioned matrices, tiny perturbations can drastically change the inverse.

Example Matrices and Invertibility Summary

Matrix A	Determinant	Inverse Exists?	Notes
[[1, 2], [3, 4]]	-2	Yes	Classic 2×2 test matrix; easy to verify manually.
[[1, 2, 3], [0, 1, 4], [5, 6, 0]]	1	Yes	Stable 3×3 example; default for demonstrating identity verification.
[[1, 2], [2, 4]]	0	No	Linearly dependent rows; calculator flags singularity immediately.

Use quick examples above to explore determinant, inverse existence, and numerical behavior.

Formula Used for Matrix Multiplicative Inverse

For a 2×2 matrix, the closed-form formula uses the adjugate and determinant. For higher orders, this calculator applies Gauss–Jordan elimination with partial pivoting on [A | I] to obtain [I | A⁻¹] reliably.

How to Use This Matrix Inverse Calculator

Select the matrix size n × n.
Enter values manually, paste from spreadsheet, or load a CSV file.
Choose decimal precision and whether to show decimals or fractions.
Optionally enable stepwise operations to display each row transformation.
Click Calculate Inverse to see A⁻¹, determinant, condition number, and verification grid.
Download the inverse as CSV or print the page to save as PDF.

Why Matrix Inverses Matter in Real Problems

Inverses help solve linear systems, analyze transformations, validate algorithms, and support control, graphics, and optimization workflows. This tool offers transparent, verifiable calculations for study, teaching, and engineering checks.

Key Features of This Matrix Inverse Calculator

Dynamic size selection, singularity detection, condition number reporting, identity verification, optional elimination steps, fractional and decimal output modes, CSV integration, and a clean white interface create a focused matrix analysis workspace.

Supported Sizes, Formats, and Export Options

Invert matrices from 2×2 up to 10×10, using integers, decimals, or simple fractions. Export the inverse as CSV for spreadsheets, and generate print-friendly pages for documentation or assignments.

Recommended Use Cases for This Tool

Ideal for students checking steps, instructors preparing solutions, engineers validating system matrices, and researchers exploring examples. Use it whenever you need reliable, explainable matrix inverses with numerical stability insights.