Cofactor Matrix Calculator

The cofactor of an element located at the i-th row and j-th column of a square matrix A is denoted as C_ij and is defined by the formula C_ij = (-1)^(i+j) * M_ij, where M_ij represents the minor of that element. The minor M_ij is the determinant of the smaller sub-matrix formed by deleting the i-th row and j-th column from A. For a 2x2 matrix, finding cofactors simply involves swapping diagonal elements and reversing specific signs. For 3x3 and 4x4 matrices, the calculator evaluates the determinants of nested 2x2 or 3x3 sub-matrices while applying the standard checkerboard sign pattern of positive and negative multipliers.

Consider a 3x3 matrix A with rows [1, 2, 1], [0, 1, 4], and [5, 6, 0]. To find the cofactor corresponding to the element in the first row and first column (a_11 = 1), we remove the first row and first column, leaving the 2x2 sub-matrix [1, 4; 6, 0]. The determinant of this sub-matrix is (1)(0) - (4)(6) = -24. Applying the sign multiplier (-1)^(1+1) = +1 gives a cofactor of -24. Repeating this process for all nine positions yields the complete cofactor matrix: [[-24, 20, -5], [6, -5, 4], [7, -4, 1]]. This resulting matrix can then be transposed to form the adjugate matrix, which is a critical step in determining the inverse of matrix A.

Always verify that your input matrix is square (having an equal number of rows and columns) before attempting to compute cofactors, as non-square matrices do not possess cofactors or determinants. Double-check your negative signs during manual checks, as sign errors are the most common source of mistakes in linear algebra. Finally, remember that the cofactor matrix must be transposed before it can be used alongside the determinant to calculate the matrix inverse.