Cofactor Matrix Calculator

The cofactor matrix replaces every entry of a square matrix with its signed minor. For an entry in row i and column j, first delete that row and column and calculate the determinant of what remains. Then apply a positive or negative sign according to the entry's checkerboard position. In a 3×3 matrix, this process creates nine 2×2 determinant calculations arranged in a new 3×3 matrix. The sign pattern begins positive in the upper-left corner and alternates across every row and column. Thus the first row uses plus, minus, plus; the second uses minus, plus, minus; and the third again uses plus, minus, plus. Applying the sign after evaluating each minor is essential. A frequent hand-calculation error is to report the matrix of minors while overlooking the four negative checkerboard positions. Cofactor matrices connect several central linear-algebra operations. Transposing the cofactor matrix produces the adjugate. If the original determinant is nonzero, dividing the adjugate by that determinant gives the inverse matrix. The dot product of any original row with the matching cofactor row yields the determinant, while pairing it with a different cofactor row yields zero. These identities provide useful checks on a result. The determinant displayed below the matrix is evaluated from the same cofactors. If it is zero, an inverse does not exist even though the cofactor matrix itself remains well-defined. Some singular matrices have nonzero cofactors, while matrices with sufficiently low rank can have an entirely zero cofactor matrix. Decimal entries are supported and all values are calculated before display rounding. Cofactors are used in solving linear systems with Cramer's rule, deriving inverse formulas, finding plane normals, studying eigenvalue polynomials, and proving determinant properties. For large numerical systems, row reduction and matrix factorizations are generally more efficient and stable. For a 3×3 exercise, however, the full cofactor matrix offers a transparent view of the structure. This calculator lists every signed result in its corresponding position, allowing you to transpose it directly when an adjugate is needed and to compare individual entries with handwritten work.