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ThmDex – An index of mathematical definitions, results, and conjectures.
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Complex matrix determinant
Definition D5718
Real matrix determinant
Formulation 0
Let
A
∈
R
N
×
N
be a
D6160: Real square matrix
.
Let
S
N
be a
D4951: Set of standard N-permutations
.
The
determinant
of
A
is the
D993: Real number
Det
A
:=
∑
π
∈
S
N
(
Sign
(
π
)
N
∏
n
=
1
A
n
,
π
(
n
)
)
Formulation 1
Let
A
∈
R
N
×
N
be a
D6160: Real square matrix
.
Let
S
N
be a
D4951: Set of standard N-permutations
.
The
determinant
of
A
is the
D993: Real number
Det
A
:=
∑
π
∈
S
N
Sign
(
π
)
A
1
,
π
(
1
)
A
2
,
π
(
2
)
A
3
,
π
(
3
)
⋯
A
N
−
1
,
π
(
N
−
1
)
A
N
,
π
(
N
)
Results
▶
R5530: Determinant of a scaled real matrix
▶
R5511: Interchanging two rows or two columns of a real square matrix switches the sign of the determinant
▶
R5519: Real arithmetic expression for the determinant of a 2-by-2 real square matrix
▶
R5513: Real matrix determinant is homogeneous with respect to multiplying a row or a column by a constant
▶
R5510: Real square matrix which has a zero column or a zero row has determinant zero