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.