The inverse of a matrix undoes what the matrix does — multiplying a matrix by its inverse always returns the identity matrix. This calculator finds A⁻¹ for 2×2 and 3×3 matrices using the adjugate method, a formula-driven approach that mirrors how the calculation is taught in a linear algebra course.
How the adjugate method works
The adjugate method builds the inverse from three ingredients: the determinant, the cofactor matrix, and the adjugate. Each cofactor is the determinant of the smaller matrix left over after deleting a row and column, multiplied by an alternating sign (+, −, +, − in a checkerboard pattern). Collecting every cofactor into a matrix and transposing it produces the adjugate. Dividing the adjugate by the determinant gives A⁻¹.
For a 2×2 matrix this reduces to a well-known shortcut: swap the two diagonal entries, negate the two off-diagonal entries, then divide everything by ad − bc. For 3×3 matrices the calculator expands the determinant along the first row and computes all nine cofactors explicitly, which the Inverse tab shows step by step.
Inputs and what they mean
Choose a matrix size (2×2 or 3×3), then fill in the grid with any real numbers — positive, negative, or decimal. There's no unit conversion involved; the calculator works on the numbers exactly as entered. Use the preset buttons to load the identity matrix, a worked example, or a deliberately singular matrix to see how the calculator handles each case.
Limits and edge cases
A matrix only has an inverse when its determinant is nonzero. If det(A) = 0, the matrix is singular and the calculator explains that no inverse exists rather than dividing by zero. This calculator supports 2×2 and 3×3 matrices only — for larger sizes, or for Gauss-Jordan row-reduction steps instead of the adjugate method, use the general-purpose Matrix Calculator, which also covers addition, multiplication, RREF, LU decomposition, and eigenvalues.