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# compound matrix

Suppose that $A$ is an $m\times n$ matrix with entries from a field $F$ and
$1\leq r\leq\min(m,n)$. The $r^{{th}}$ *compound matrix* or
$r^{{th}}$ *adjugate* of $A$ is the
$\binom{m}{r}\times\binom{n}{r}$
matrix whose entries are $\det A[\alpha,\beta])$,
$\alpha\in Q_{{r,m}}$ and $\beta\in Q_{{r,n}}$, arranged in lexicographic order and we use submatrix notation.
The notation for this matrix is $C_{r}(A)$.

1. 2. If $A$ is nonsingular, the $C_{r}(A)^{{-1}}=C_{r}(A^{{-1}})$.

3. If $A$ has complex entries, then $C_{r}(A^{*})=(C_{r}(A))^{*}$.

4. $C_{r}(A^{T})=(C_{r}(A))^{T}$

5. $C_{r}(\overline{A})=\overline{C_{r}(A)}$

6. For any $k\in F$ $C_{r}(kA)=k^{r}C_{r}(A)$

7. $C_{r}(I_{n})=I_{{\binom{n}{r}}}$

8. $\det(C_{r}(A))=\det(A)^{{\binom{n-1}{r-1}}}$ (Sylvester — Franke theorem)

## Mathematics Subject Classification

15-00*no label found*

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