Inverse Matrix Calculator

The inverse of a square matrix A is the matrix A⁻¹ that satisfies A·A⁻¹ = A⁻¹·A = I, where I is the identity matrix. It plays the same role for matrices that 1/a plays for numbers.

Geometrically, a 2×2 matrix moves every point of the plane: it can stretch, rotate, shear, or reflect. The inverse matrix is the transformation that moves every point back to where it started.

1. Square only: A must have the same number of rows and columns.
2. Non-zero determinant: det(A) ≠ 0, otherwise the transformation flattens space and cannot be reversed.
3. Unique: if an inverse exists, there is exactly one.
4. Determinants multiply: det(A⁻¹) = 1/det(A), so areas scaled by A are unscaled by A⁻¹.

For A = [[a, b], [c, d]], the inverse is A⁻¹ = (1/(ad − bc))·[[d, −b], [−c, a]].

1. Find the determinant ad − bc. If it is 0, stop: there is no inverse.
2. Swap the two entries on the main diagonal, a and d.
3. Negate the two off-diagonal entries, b and c.
4. Divide every entry by the determinant.

Example: for [[4, 7], [2, 6]], det = 24 − 14 = 10, so A⁻¹ = (1/10)·[[6, −7], [−2, 4]] = [[0.6, −0.7], [−0.2, 0.4]].

A⁻¹ = (1/det A)·adj(A), where adj(A) is the transpose of the cofactor matrix. This is the 3×3 version of the swap-and-negate rule.

1. Minors: for each entry, delete its row and column and take the determinant of the 2×2 that is left.
2. Cofactors: apply the checkerboard of signs + − + / − + − / + − + to the minors.
3. Adjugate: transpose the cofactor matrix (rows become columns).
4. Divide: multiply the adjugate by 1/det A.

The adjugate method needs 9 small determinants for a 3×3 but grows quickly for larger matrices, which is why calculators and computers use Gauss-Jordan elimination for 4×4 and up.

A matrix is not invertible (singular) exactly when its determinant is 0. All of these statements are equivalent for a square matrix:

• det(A) = 0.
• Some row is a combination of the other rows (the rows are linearly dependent).
• Ax = 0 has a solution other than x = 0.
• Gauss-Jordan elimination produces a row of zeros on the left side.
• A flattens the plane onto a line (or space onto a plane), so information is lost.

Nearly singular matrices are also a problem in practice: when det(A) is tiny, small rounding errors in the entries cause huge errors in A⁻¹.

Inverses follow a small set of rules that make algebra with matrices possible. For invertible n×n matrices A and B and a non-zero number k:

• (A⁻¹)⁻¹ = A: undoing the undo gives the original.
• (AB)⁻¹ = B⁻¹A⁻¹: the order reverses.
• (Aᵀ)⁻¹ = (A⁻¹)ᵀ: transpose and inverse can be swapped.
• (kA)⁻¹ = (1/k)·A⁻¹: scalars come out as reciprocals.
• det(A⁻¹) = 1/det(A).

Note that (A + B)⁻¹ is generally not A⁻¹ + B⁻¹. There is no simple sum rule.

If Ax = b and A is invertible, the unique solution is x = A⁻¹b. Multiply both sides on the left by A⁻¹ and use A⁻¹A = I.

Each equation of a 2×2 system is a line in the plane, and the solution is the point where the two lines cross. If det A = 0 the lines are parallel (no solution) or identical (infinitely many solutions), which is exactly when A⁻¹ does not exist.

Computing A⁻¹ once is useful when you must solve Ax = b for many different right-hand sides b, as in circuit analysis, input-output economic models, and 3D graphics, where the same camera matrix is inverted once and applied to millions of points.