• (Sylvester-Franke Theorem)

[1]

As in [2], introduce the sign matrix diagonal matrix with entries alternating with . And the reversal matrix with 1's on the antidiagonal and zeros elsewhere.


  • (see below)

Compound matrices and adjugates

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[See [3] for a classical discussion related to this section.]

Recall the adjugate matrix is the transpose of the matrix of cofactors, signed minors complementary to single entries. Then we can write

  (1)


with   denoting transpose.

The basic property of the adjugate is the relation

 ,

hence   while

  (2)

Comparing these and using the Sylvester-Franke theorem yields the identity

  •  

Jacobi's Theorem on the Adjugate

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Jacobi's Theorem extends (1) to higher-order minors [2]:

 

expressing minors of the adjugate in terms of complementary signed minors of the original matrix.

Substituting into the previous identity and going back to (2) yields

  and hence the formula for the inverse of the compound matrix given above.


  1. ^ Tornheim, Leonard (1952). "The Sylvester-Franke Theorem". The American Mathematical Monthly. 59 (6): 389. doi:10.2307/2306811. ISSN 0002-9890.
  2. ^ a b Nambiar, K.K.; Sreevalsan, S. (2001). "Compound matrices and three celebrated theorems". Mathematical and Computer Modelling. 34 (3–4): 251–255. doi:10.1016/S0895-7177(01)00058-9. ISSN 0895-7177.
  3. ^ Price, G. B. (1947). "Some Identities in the Theory of Determinants". The American Mathematical Monthly. 54 (2): 75. doi:10.2307/2304856. ISSN 0002-9890.