In mathematics, the kth compound matrix Ck(A) of an m × n matrix A is the ( ( m k ) − 1 ) × ( ( n k ) − 1 ) {\displaystyle \left({\binom {m}{k}}-1\right)\times \left({\binom {n}{k}}-1\right)} matrix formed from the determinants of all k × k submatrices of A arranged with the submatrix index sets in lexicographic order.
Adjugate matrix: the (n − 1)th compound matrix of an n × n matrix
matrix formed from the determinants of all k × k submatrices of A arranged with the submatrix index sets in lexicographic order.
![{\displaystyle {\begin{aligned}C_{1}(A)&=A\\[6pt]C_{n}(A)&=\det(A){\text{ if }}A{\text{ is }}n\times n\\[6pt]C_{k}(AB)&=C_{k}(A)C_{k}(B)\\[6pt]C_{k}(aX)&=akC_{k}(X)\\[6pt]C_{k}(I)&=I\\[6pt]C_{k}(A^{H})&=C_{k}(A)^{H}\\[6pt]C_{k}(A^{T})&=C_{k}(A)^{T}\\[6pt]C_{k}(A^{-1})&=C_{k}(A)^{-1}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9b818cbc7faa8325538723c6ea090e6b16edf2e)
