Compound matrix

In mathematics, the kth compound matrix C_k(A) of an m × n matrix A is the ${\displaystyle \left({\binom {m}{k}}-1\right)\times \left({\binom {n}{k}}-1\right)}$ matrix formed from the determinants of all k × k submatrices of A arranged with the submatrix index sets in lexicographic order.

${\displaystyle {\begin{aligned}C_{1}(A)&=A\\[6pt]C_{n}(A)&=\det(A){\text{ if }}A{\text{ is }}n\times n\\[6pt]C_{k}(AB)&=C_{k}(A)C_{k}(B)\\[6pt]C_{k}(aX)&=akC_{k}(X)\\[6pt]C_{k}(I)&=I\\[6pt]C_{k}(A^{H})&=C_{k}(A)^{H}\\[6pt]C_{k}(A^{T})&=C_{k}(A)^{T}\\[6pt]C_{k}(A^{-1})&=C_{k}(A)^{-1}\end{aligned}}}$

See also

Adjugate matrix: the (n − 1)th compound matrix of an n × n matrix

To efficiently calculate compound matrices see: "Compound matrices: properties, numerical issues and analytical computations" - Christos Kravvaritis · Marilena Mitrouli - DOI 10.1007/s11075-008-9222-7