Inverse Matrix Calculator
This tool helps you find the inverse of any square matrix using two well-known methods: Adjoint (Cofactor) method and Gauss-Jordan elimination.
Simply enter your matrix, select the method, and get the full step-by-step solution instantly.
What Is a Matrix Inverse?
The inverse of a matrix A is another matrix A-1 such that:
A · A⁻¹ = I
where I is the identity matrix. Only square matrices (same number of rows and columns) with a non-zero determinant have an inverse.
Why Calculate a Matrix Inverse?
- Solving systems of linear equations
- Computer graphics and 3D transformations
- Cryptography
- Control systems and engineering
Knowing how to compute the inverse is fundamental in linear algebra.
Methods We Use
1. Adjoint (Cofactor) Method
- Compute the determinant
- Find the matrix of cofactors
- Take the transpose (adjugate)
- Divide by the determinant
This method is suitable for small matrices (e.g., 2×2, 3×3).
2. Gauss-Jordan Elimination
- Augment the original matrix with the identity matrix
- Use row operations to reduce it to the identity matrix
- The transformed identity becomes the inverse
This method is more practical for larger matrices.
Example: Inverse of a 3×3 Matrix
Given the matrix:
A = [ [2, 1, 3],
[0, 1, 4],
[5, 2, 0] ]
You can compute the inverse using either method, and we will walk you through every step.
When Is the Inverse Not Defined?
- It’s not square (e.g., 2×3, 4×2)
- Its determinant is zero (also called a singular matrix)
Our calculator will automatically detect such cases and alert you.
Tips for Best Results
- Use fractions instead of decimals for exact values
- Double-check: A · A⁻¹ = I
- Prefer Gauss-Jordan for matrices larger than 3×3