compound matrix
Suppose that $A$ is an $m\times n$ matrix with entries from a field $F$ and $1\le r\le \mathrm{min}(m,n)$. The ${r}^{th}$ compound matrix or ${r}^{th}$ of $A$ is the $\left(\genfrac{}{}{0pt}{}{m}{r}\right)\times \left(\genfrac{}{}{0pt}{}{n}{r}\right)$ matrix whose entries are $detA[\alpha ,\beta ])$, $\alpha \in {Q}_{r,m}$ 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.
${C}_{r}(AB)={C}_{r}(A){C}_{r}(B)$ when $r$ is less than or equal to the number of rows or columns of $A$ and $B$

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.
${C}_{r}({I}_{n})={I}_{\left(\genfrac{}{}{0pt}{}{n}{r}\right)}$

8.
$det({C}_{r}(A))=det{(A)}^{\left(\genfrac{}{}{0pt}{}{n1}{r1}\right)}$ (Sylvester — Franke theorem^{})
Title  compound matrix 

Canonical name  CompoundMatrix 
Date of creation  20130322 16:13:39 
Last modified on  20130322 16:13:39 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  9 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 1500 
Defines  rth adjugate 
Defines  Sylvester Franke theorem 