Formal definition

If A is a square matrix, then the minor of its entry aij, also known as the i,j, or (i,j), or (i,j)th minor of A, is denoted by Mij and is defined to be the determinant of the submatrix obtained by removing from A its i-th row and j-th column.
The number
is denoted by Cij and is called the cofactor of aij, also referred to as the i,j, (i,j) or (i,j)th cofactor of A.

Example

Given the matrix
suppose we wish to find the cofactor C23. The minor M23 is the determinant of the above matrix with row 2 and column 3 removed.
yields Using the given definition it follows that
Note: the vertical lines are an equivalent notation for det(matrix)

Matrix of cofactors


The matrix of cofactors for an matrix A is the matrix whose (i,j) entry is the cofactor Cij of A. For instance, if A is
the cofactor matrix of A is
where Cij is the cofactor of aij.

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