Matrix Inverse Calculator

This calculator uses the adjugate formula: A inverse equals adj(A) divided by det(A). It first computes the determinant, then the cofactor matrix, transposes the cofactor matrix to get the adjugate, and divides every entry by the determinant. The result is shown alongside the determinant and the adjugate so you can follow each step.

A square matrix is invertible only when its determinant is not zero. If the determinant is zero, the matrix is called singular and no inverse exists, because dividing the adjugate by zero is undefined. This happens when the rows or columns are linearly dependent, for example when one row is a multiple of another or a full row of zeros appears. The calculator flags this case clearly instead of returning numbers.

For A equal to [[a, b], [c, d]], the determinant is ad minus bc. The inverse is 1 over (ad minus bc) times [[d, -b], [-c, a]]. In other words you swap a and d, negate b and c, and divide every entry by the determinant. The calculator applies exactly this rule for the 2x2 case.

The adjugate, also called the classical adjoint, is the transpose of the cofactor matrix. Each cofactor is the signed minor of an entry, where the sign alternates as (-1) raised to (i plus j). Dividing the adjugate by the determinant gives the inverse, so the adjugate panel shows the numerators before scaling.

This calculator inverts 2x2 and 3x3 square matrices, which are the sizes most common in coursework and quick checks. Pick the size from the dropdown, fill in every cell, and the inverse updates automatically. Larger matrices need Gaussian elimination or LU decomposition, which are better suited to a dedicated tool.

Inverse entries are exact fractions of the form cofactor over determinant, so they are only whole numbers when the determinant evenly divides each cofactor. When it does not, you get decimals, which the calculator rounds to six places. The inverse is made of integers exactly when the determinant is 1 or -1 for an integer matrix, which is why the integer inverse flag is shown.