Inverse Matrix

The inverse matrix calculator finds the inverse by Gauss–Jordan elimination (up to 6×6) or by the adjugate (cofactor) method (up to 4×4), with a detailed solution and the check A · A⁻¹ = E. If the determinant is zero, the calculator reports that the inverse does not exist.

What is an inverse matrix

The inverse A⁻¹ is the matrix with A · A⁻¹ = A⁻¹ · A = E. It exists only for a square matrix with a non-zero determinant.

Gauss–Jordan elimination

Write the augmented matrix [A | E] and use row operations to turn the left part into the identity. The right part becomes A⁻¹: [A | E] → [E | A⁻¹].

Adjugate method

A⁻¹ = (1 / det A) · adj A,   adj A = (A_ij)^T,   A_ij = (−1)^i+jM_ij

For a 2×2 matrix: if A = (a b; c d), then A⁻¹ = 1/(ad − bc) · (d −b; −c a).

Example. A = (2 1 1; 1 3 2; 1 0 0), det A = −1.
A⁻¹ = (0 0 1; −2 1 3; 3 −1 −5).

The inverse is used to solve matrix equations AX = B (X = A⁻¹B) and systems of linear equations.