Inverse Matrix Calculator

The inverse of a square matrix A is another matrix A⁻¹ such that A × A⁻¹ = A⁻¹ × A = I, where I is the identity matrix. Not all matrices have inverses — only square matrices with a nonzero determinant are invertible.

A square matrix has an inverse if and only if its determinant is nonzero. If det(A) = 0, the matrix is called singular and has no inverse.

For a 2×2 matrix [[a,b],[c,d]], the inverse is (1/(ad-bc)) × [[d, -b], [-c, a]], provided ad-bc ≠ 0.

For a 3×3 matrix, compute the matrix of minors, then the cofactor matrix, then transpose to get the adjugate, and finally divide by the determinant: A⁻¹ = adj(A)/det(A).

Gaussian elimination augments the matrix with the identity matrix [A|I] and performs row operations until the left side becomes I. The right side then becomes A⁻¹.

No, only square matrices can have a standard two-sided inverse. Non-square matrices can have a left inverse or right inverse (pseudoinverse) under specific conditions.

The adjugate (or classical adjoint) is the transpose of the cofactor matrix. Each cofactor is computed as (-1)^(i+j) times the minor determinant obtained by deleting row i and column j.

The determinant appears in the denominator of the inverse formula. A larger absolute determinant means the matrix transformation is less "compressed," while a zero determinant means the transformation collapses space and is irreversible.

Yes, if a matrix has an inverse, that inverse is unique. There is only one matrix A⁻¹ that satisfies A × A⁻¹ = I.

The identity matrix is its own inverse: I⁻¹ = I. Multiplying I by itself gives I.

Yes, for Ax = b, you can compute x = A⁻¹b. However, in practice, Gaussian elimination is more numerically stable and efficient than computing the full inverse.

The Moore-Penrose pseudoinverse extends the concept of matrix inverse to non-square or singular matrices. It is denoted A⁺ and provides the best approximate solution to Ax = b.

For an n×n matrix, computing the inverse requires approximately O(n³) operations. For very large matrices, iterative methods or direct solvers are preferred over explicit inversion.

The condition number κ(A) = ||A|| × ||A⁻¹|| measures how sensitive the inverse is to small changes in the input. A high condition number means the matrix is nearly singular and its inverse is numerically unreliable.

Yes, but in reverse order: (AB)⁻¹ = B⁻¹A⁻¹. This is because (B⁻¹A⁻¹)(AB) = B⁻¹(A⁻¹A)B = B⁻¹IB = B⁻¹B = I.

The inverse of the transpose equals the transpose of the inverse: (A⁻¹)ᵀ = (Aᵀ)⁻¹. This means you can compute the transpose first and then invert, or vice versa.

The inverse of a diagonal matrix is simply the diagonal matrix with each nonzero diagonal element replaced by its reciprocal. If any diagonal element is zero, the matrix is singular.

Yes, a matrix satisfying A² = I is called an involution. Examples include reflection matrices and the Pauli spin matrices in quantum mechanics.

The inverse of a rotation matrix is its transpose (equivalently, the rotation by the negative angle). Rotation matrices are orthogonal, so R⁻¹ = Rᵀ.

Multiply the original matrix by the computed inverse from both sides: A × A⁻¹ should equal I and A⁻¹ × A should also equal I.

The Inverse Matrix Calculator uses standard mathematical formulas validated against authoritative references. The specific formula is displayed in the calculator interface with a detailed explanation of each variable.

Each calculator includes a step-by-step worked example showing exactly how the formula is applied. You can follow these steps with pen and paper to verify any result.

The Inverse Matrix Calculator accepts numeric inputs including integers and decimals. Invalid inputs (letters, special characters) are rejected with clear error messages.