Determinante (rekursive Definition)
Es sei
K
{\displaystyle {}K}
Körper und sei
M
=
(
a
i
j
)
i
j
{\displaystyle {}M={\left(a_{ij}\right)}_{ij}}
n
×
n
{\displaystyle {}n\times n}
Matrix
über
K
{\displaystyle {}K}
i
∈
{
1
,
…
,
n
}
{\displaystyle {}i\in {\{1,\ldots ,n\}}}
M
i
{\displaystyle {}M_{i}}
(
n
−
1
)
×
(
n
−
1
)
{\displaystyle {}(n-1)\times (n-1)}
M
{\displaystyle {}M}
i
{\displaystyle {}i}
Determinante
M
{\displaystyle {}M}
det
M
=
{
a
11
,
falls
n
=
1
,
∑
i
=
1
n
(
−
1
)
i
+
1
a
i
1
det
M
i
für
n
≥
2
.
{\displaystyle {}\det M={\begin{cases}a_{11}\,,&{\text{falls }}n=1\,,\\\sum _{i=1}^{n}(-1)^{i+1}a_{i1}\det M_{i}&{\text{ für }}n\geq 2\,.\end{cases}}\,}
