cofactors

Every element of a matrix has a cofactor, which is obtained as follows:

1. Locate the row and column which contain the element. 2. Remove that row and column from the original matrix. 3. Take the determinant of the resulting matrix.

The determinant is the cofactor.

More specifically:

If the matrix is M and element is in row i and column j, then matrix you get when you remove row i and column j is denoted M_ij.

The i,j cofactor is det(M_ij).

For example for the matrix

[ [2, 3, -4], [5, -1, -3], [-2, 7, 6] ]

M_12 would be the matrix obtained by eliminating row 1 and column 2. Row 1 is [2, 3, -4] and column 2 consists of the 'middle' column, containing numbers 3, -1 and 7. Thus

M_12 = [ [5, -3], [-2, 6] ]

and the 1,2 cofactor is

det( [ [5, -3], [-2, 6] ] ) = 5 * 6 - (-3) * (-2) = 30 - 6 = 24.