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 * $$\det(C_k(A))=(\det A)^{\binom{n-1}{k-1}}$$ (Sylvester-Franke Theorem)

As in, introduce the sign matrix $$S=$$ diagonal matrix with entries alternating $$\pm1$$ with $$S_{11}=1$$. And the reversal matrix $$J$$ with 1's on the antidiagonal and zeros elsewhere.


 * $$C_k(A)^{-1}=\det(A)^{-1}J(C_{n-k}(SAS))^TJ$$ (see below)

Compound matrices and adjugates
[See 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

with $$T$$ denoting transpose.

The basic property of the adjugate is the relation

$$A\operatorname{adj}(A)=\det(A) I$$,

hence $$C_k(A)C_k(\operatorname{adj}(A))=\det(A)^k I$$ while

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


 * $$\operatorname{adj}(C_k(A))=\det(A)^{\binom{n-1}{k-1}-k} C_k(\operatorname{adj}(A))$$

Jacobi's Theorem on the Adjugate
Jacobi's Theorem extends ($$) to higher-order minors :

$$C_k(\operatorname{adj}(A))=(\det A)^{k-1} J(C_{n-k}(SAS))^TJ$$ expressing minors of the adjugate in terms of complementary signed minors of the original matrix.

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

$$C_k(A)\, J(C_{n-k}(SAS))^TJ=\det(A) I$$ and hence the formula for the inverse of the compound matrix given above.