Sylvester's determinant identity

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Not to be confused with the Weinstein–Aronszajn identity, which is sometimes attributed to Sylvester.

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 × n matrix , let denote its determinant. Choose a pair

of -element ordered subsets of , . Let denote the submatrix of obtained by deleting the rows in and the columns in . Define the auxiliary matrix whose elements are equal to the following determinants

where , denote the element subsets of and obtained by deleting the elements and , respectively. Then the following is Sylvester's determinantal identity (Sylvester, 1851)

When , this is the Desnanot-Jacobi identity (Jacobi, 1851)


  1. ^ Sylvester, James Joseph (1851). "On the relation between the minor determinants of linearly equivalent quadratic functions". Philosophical Magazine. 1: 295–305.
    Cited in Akritas, A. G.; Akritas, E. K.; Malaschonok, G. I. (1996). "Various proofs of Sylvester's (determinant) identity". Mathematics and Computers in Simulation. 42 (4–6): 585. doi:10.1016/S0378-4754(96)00035-3.