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ECON 401: Determinant and Inverse of a Matrice
Siyang Xiong
Rice University
August 25, 2011
Xiong (Rice University)
ECON 401
August 25, 2011
1 / 20
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n
equations of
n
variable
a
11
x
1
C
a
12
x
2
C
...
C
a
1
n
x
n
D
b
1
(1)
a
21
x
1
C
a
22
x
2
C
...
C
a
2
n
x
n
D
b
2
...
a
n
1
x
1
C
a
n
2
x
2
C
...
C
a
nn
x
n
D
b
n
(3) can be represented as follows:
2
4
a
11
...
a
1
n
...
...
...
a
n
1
...
a
nn
3
5
2
4
x
1
...
x
n
3
5
D
2
4
b
1
...
b
n
3
5
Xiong (Rice University)
ECON 401
August 25, 2011
2 / 20
we use
j
A
j
to denote the determinant of a
n
n
matrice
A
for a 1
1 matrice
A
D
[
a
]
[
a
]
D
a
.
Given a
n
n
matrice
A
,
let
A
.
i
,
j
/
denote the matrice resulted by deleting the
i
th row and
j
th
column of
A
.
C
ij
D
.
±
1
/
i
C
j
j
A
.
i
,
j
/
j
is called the (i,j)th cofactor of
A
;
then,
j
A
j
D
a
i
1
.
±
1
/
i
C
1
j
A
.
i
,
j
/
jC
...
C
a
in
C
in
D
a
1
j
C
1
j
C
...
C
a
nj
C
nj
Xiong (Rice University)
ECON 401
August 25, 2011
3 / 20
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±
a
b
c
d
D
a
.
±
1
/
1
C
1
[
d
]
C
b
.
±
1
/
2
C
1
[
c
]
D
ad
±
bc
.
±
a
b
c
d
D
b
.
±
1
/
1
C
2
[
c
]
C
d
.
±
1
/
2
C
2
[
a
]
D ±
bc
C
ad
.
Xiong (Rice University)
ECON 401
August 25, 2011
4 / 20
2
4
1
1
6
2
4
3
2
4
5
3
5
D
1
.
±
1
/
1
C
1
±
4
3
4
5
C
1
.
±
1
/
1
C
2
±
2
3
2
5
C
6
.
±
1
/
1
C
3
±
2
4
2
4
D
[
4
.
±
5
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This note was uploaded on 02/09/2012 for the course ECON 401 taught by Professor Siyang during the Spring '11 term at Rice.
 Spring '11
 Siyang
 Economics

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