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

1. Compute the determinant
2. Find the matrix of cofactors
3. Take the transpose (adjugate)
4. 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