In matrix theory, Sylvester's determinant identity is an identity useful for evaluating certain types of determinants. It is named after James Joseph Sylvester, who stated this identity without proof in 1851.[1]
Given an n-by-n matrix , let denote its determinant. Choose a pair
of m-element ordered subsets of , where m ≤ n. Let denote the (n−m)-by-(n−m) submatrix of obtained by deleting the rows in and the columns in . Define the auxiliary m-by-m matrix whose elements are equal to the following determinants
where , denote the m−1 element subsets of and obtained by deleting the elements and , respectively. Then the following is Sylvester's determinantal identity (Sylvester, 1851):
When m = 2, this is the Desnanot-Jacobi identity (Jacobi, 1851).