Inverse Matrix Calculator

How to use

An inverse matrix reverses an invertible linear transformation. This calculator handles numerical square matrices of size 2 × 2 or 3 × 3, suitable for checking small algebra problems and coordinate transformations. Choose the size before entering entries. Each label aij identifies row i and column j, starting at 1. Keep the original row order: transposing or swapping entries changes the matrix and usually changes its inverse.

Replace the editable example values with your own data, choose the comparison or dimension where available, and press Calculate. Enter submits the same form. The result is confirmed only after this action. Editing a number or changing a selection removes the previous result and disables Copy result, so an old answer cannot be mistaken for the current calculation. Clear empties the number fields while keeping the selected mode. After a successful calculation, Copy result copies the input and the numerical output as plain text. If clipboard access is blocked, select the visible text instead. Pale values are a runnable example. Focusing a number field clears all example numbers once. Your later entries are kept.

How it works and reading the result

For a 2 × 2 matrix with rows (a, b) and (c, d), the determinant is ad − bc and the inverse has rows (d, −b) and (−c, a), divided by that determinant. For 3 × 3, cofactors form the adjugate, which is divided by the determinant. The calculator converts the entered decimals to a common integer scale and computes determinant and cofactors exactly before producing numerical output. A small nonzero determinant is not automatically treated as zero.

Read the three aligned grids as the input matrix A, its inverse A⁻¹, and the product A × A⁻¹. The identity matrix has ones on the main diagonal and zeros elsewhere. The maximum absolute difference from identity is shown separately, using the unrounded inverse. The condition estimate κ∞ = ‖A‖∞ ‖A⁻¹‖∞ measures sensitivity to input perturbations; each infinity norm is the largest absolute row sum. A large condition estimate warns that small input or rounding changes can substantially affect the answer.

Examples

Example 1: choose 2 × 2 and enter rows (4, 7) and (2, 6). The determinant is 10 and the inverse has rows (0.6, −0.7) and (−0.2, 0.4). Multiplication gives rows (1, 0) and (0, 1). For instance, the first entry is 4 × 0.6 + 7 × (−0.2) = 1.

Example 2: choose 3 × 3 and enter the diagonal matrix with diagonal entries 2, 4 and 5, setting all other entries to zero. Its inverse has diagonal entries 0.5, 0.25 and 0.2, the determinant is 40, and the product is the 3 × 3 identity. By contrast, rows (1, 2) and (2, 4) form a singular 2 × 2 matrix: no inverse exists.

Limits and troubleshooting

Entries must be zero or have magnitudes from 1e−12 to 1e12, using decimal points or scientific notation. Fractions, symbolic entries, nonsquare matrices and sizes above 3 are excluded. Every active entry must be filled; blank does not mean zero. The singular decision applies to the exact decimal strings you entered, while the inverse and product use floating-point output. A warning appears for κ∞ above 1e10 or an identity error above 1e−8. A well-looking rounded product alone does not guarantee accuracy for an ill-conditioned matrix. Keep the original input and error with copied results.

Use at most 15 significant input digits.

Frequently asked questions