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. 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 When m = 2, this is the Desnanot-Jacobi identity (Jacobi, 1851).