Matrix Inverse Calculator

Calculate the inverse of a matrix. Free online matrix inverse calculator.

The Matrix Inverse Calculator finds the inverse A⁻¹ of a square matrix where A × A⁻¹ = I (identity matrix). The inverse exists only when the determinant is non-zero. Enter your matrix to find its inverse with step-by-step solution, essential for solving systems of equations and linear algebra applications.

Common use cases

  • Solving systems of equations
  • Cryptography decryption
  • Computer graphics transformations
  • Regression analysis
  • Network flow calculations

How to use

  1. Select matrix size (2×2, 3×3, etc.)
  2. Enter the matrix elements
  3. Click calculate for the inverse
  4. View the calculation method
  5. Verify A × A⁻¹ = I

FAQ

When does a matrix have an inverse?

When its determinant ≠ 0. Such matrices are called 'non-singular' or 'invertible'.

How is inverse calculated?

For 2×2: swap diagonals, negate off-diagonals, divide by determinant. Larger matrices use row reduction or adjugate method.

What if determinant is 0?

The matrix is singular and has no inverse. The system of equations has either no solution or infinitely many.

What are properties of inverse?

(A⁻¹)⁻¹ = A, (AB)⁻¹ = B⁻¹A⁻¹, and (Aᵀ)⁻¹ = (A⁻¹)ᵀ.