Inverse Matrix Calculator

Reviewed by CalcMulti Editorial Team·Last updated: March 2026·← Algebra Hub

The inverse of a matrix A, written A⁻¹, is the matrix that satisfies A · A⁻¹ = A⁻¹ · A = I, where I is the identity matrix. Not every matrix has an inverse — a matrix is invertible only when its determinant is non-zero. Invertible matrices are central to solving systems of linear equations of the form Ax = b, since the solution is simply x = A⁻¹b.

This calculator finds the inverse of a 2×2 or 3×3 matrix using the adjugate (classical adjoint) method: A⁻¹ = (1/det(A)) · adj(A). For each entry the calculator displays the cofactor matrix, the adjugate, the determinant, and the final inverse — so you can follow every arithmetic step.

The determinant is the gatekeeper of invertibility. For a 2×2 matrix [[a, b], [c, d]], the determinant is ad − bc. If this equals zero, the two row vectors are linearly dependent (one is a scalar multiple of the other), meaning the matrix collapses 2D space onto a line — a transformation that cannot be reversed. For 3×3 matrices, a zero determinant means the three rows lie in a common plane rather than spanning 3D space. Geometrically, the absolute value of the determinant is the area (2×2) or volume (3×3) scaling factor of the linear transformation represented by A.

For a 2×2 matrix, the inverse formula is elegant: swap the main diagonal entries (a and d), negate the off-diagonal entries (b and c), and divide by the determinant. For a 3×3 matrix, the process requires computing nine cofactors (each the signed determinant of a 2×2 submatrix), assembling them into the cofactor matrix, transposing it to form the adjugate, and dividing by the determinant. While tedious by hand, understanding each step — cofactors, adjugate, scaling — builds genuine intuition for how linear transformations can be reversed.

Matrix inverses appear throughout applied mathematics and engineering. Solving a system of n linear equations with n unknowns reduces to computing A⁻¹b. In computer graphics, inverse transformation matrices are used to undo rotations, scalings, and translations of 3D objects. In statistics, the matrix (XᵀX)⁻¹ appears in the formula for ordinary least squares regression coefficients. In control theory, invertible state-transition matrices allow engineers to reconstruct initial conditions from observed outputs. Whenever a linear transformation must be reversed or a linear system must be solved, the matrix inverse is the fundamental tool.

Complete 2×2 worked example: find A⁻¹ for A = [[4, 7], [2, 6]]. Step 1 — determinant: det(A) = (4)(6) − (7)(2) = 24 − 14 = 10. Step 2 — since det ≠ 0, A is invertible. Step 3 — swap main diagonal and negate off-diagonal: [[6, −7], [−2, 4]]. Step 4 — divide by det: A⁻¹ = (1/10)·[[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]]. Verification: A · A⁻¹ = [[4·0.6 + 7·(−0.2), 4·(−0.7) + 7·0.4], [2·0.6 + 6·(−0.2), 2·(−0.7) + 6·0.4]] = [[2.4−1.4, −2.8+2.8], [1.2−1.2, −1.4+2.4]] = [[1, 0], [0, 1]] = I ✓.

For 3×3 matrices, the cofactor method systematically handles the greater complexity. Given A = [[1, 2, 3], [0, 4, 5], [1, 0, 6]]: det(A) = 1(24−0) − 2(0−5) + 3(0−4) = 24 + 10 − 12 = 22. Each of the nine cofactors requires computing a 2×2 determinant with the appropriate sign: C₁₁ = +(4·6 − 5·0) = 24, C₁₂ = −(0·6 − 5·1) = 5, C₁₃ = +(0·0 − 4·1) = −4, and so on. Arrange the cofactors in a 3×3 matrix, transpose it to form adj(A), then divide by det(A). While the arithmetic is extensive, each step is deterministic — this calculator performs all nine cofactor calculations automatically and shows the full adjugate before dividing.

Properties of matrix inverses that simplify multi-step calculations: (1) (A⁻¹)⁻¹ = A — the inverse of the inverse is the original. (2) (AB)⁻¹ = B⁻¹A⁻¹ — the inverse of a product reverses the order of individual inverses (analogous to putting on and taking off socks and shoes). (3) (Aᵀ)⁻¹ = (A⁻¹)ᵀ — transpose and inverse commute. (4) det(A⁻¹) = 1/det(A) — the determinant of the inverse is the reciprocal. (5) If A is symmetric (A = Aᵀ), then A⁻¹ is also symmetric. These properties are frequently tested in linear algebra courses and are essential for deriving efficient computational algorithms.