In mathematics, the kth compound matrix (sometimes referred to as the kth multiplicative compound matrix)
C
k
(
A
)
, of an
m
×
n
matrix A is the
(
m
k
)
×
(
n
k
)
matrix formed from the determinants of all
k
×
k
submatrices of A, i.e., all
k
×
k
minors, arranged with the submatrix index sets in lexicographic order. The following properties hold:
C
1
(
A
)
=
A
C
n
(
A
)
=
det
(
A
)
if
A
is
n
×
n
C
k
(
A
B
)
=
C
k
(
A
)
C
k
(
B
)
C
k
(
a
X
)
=
a
k
C
k
(
X
)
For
n
×
n
identity
I
,
C
k
(
I
)
=
I
,
the
(
n
k
)
×
(
n
k
)
identity
C
k
(
A
T
)
=
C
k
(
A
)
T
,
over any field
C
k
(
A
∗
)
=
C
k
(
A
)
∗
,
over
C
C
k
(
A
−
1
)
=
C
k
(
A
)
−
1
,
for
n
×
n
,
invertible
A
If
A
is viewed as the matrix of an operator in a basis
(
e
1
,
…
,
e
n
)
then the compound matrix
C
k
(
A
)
is the matrix of the
k
-th exterior power
A
∧
k
in the basis
(
e
i
1
∧
⋯
∧
e
i
k
)
i
1
<
⋯
<
i
k
. In this formulation, the multiplicativity property
C
k
(
A
B
)
=
C
k
(
A
)
C
k
(
B
)
is equivalent to the functoriality of the exterior power.
