The Determinant of Square Matrix
description
Transcript of The Determinant of Square Matrix
-
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
fWhat you should learn -
2x2 -
-
fWhy you should learn it
2. The Determinant of Square Matrix
(determinant) n nn A (scale factor) A (multilinear algebra) (substitution rule) n nn (commutative ring)
1. 2. A A det(A) |A|
2 X 2 2 x 2
d
a bA
c
=
A
det( ) |A| a b
A ad bcc d
= = =
3 X 3 1 2 2
A 3 x 3 A = 1 1 1
2 2 3
3 3 3
a b c
a b c
a b c
A det (A) 1 1 1 1 12 2 3 2 2
3 3 3 3 3
a b c a b
a b c a b
a b c a b
1 2 3 1 2 3 1 2 3 3 2 1 3 2 1 3 2 1
det( ) ( ) ( )A a b c b c a c a b a b c b c a c a b= + + + +
-
| Page 2
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
1 A = [ 5 ] B = [ -8 ] det( )A det( )B
2 4 3 4 2 1 5, ,2 1 2 1 2 1
A B C = = =
det( )det( )det( )A B C
3 1 2 3
0 4 1
1 2 0
A
= det( )A
4 2 1 0
3 4 5
9 8 7
A
= det( )A
-
| Page 3
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.1 Minor Cofactor 2.1.1 Minor A n x n n > 2 1.
ija
ijM
2. ija i j
A
5 2 1 0
3 4 5
9 8 7
A =
A
2.1.2 Cofactor A n x n n > 2 1.
ija
ijC
2. ija ( 1)i j
ijM+
6 2 0 1
3 1 2
4 5 6
A
=
A
-
| Page 4
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
Minor Cofactor 2.1.1 [ ]
ij n nA a = ija n > 2
1. 1 1 2 2
det( ) ... ,i i i i in in
A a C a C a C i= + + + i 2.
1 1 2 2det( ) ... ,
j j j j nj njA a C a C a C j= + + + j
7 A = 4 1 12 1 2
3 5 2
det (A)
8 A = 1 1 1 2
1 2 2 1
4 3 0 1
3 0 2 1
det (A)
-
| Page 5
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.1.3 [ ]
ij n nA a = ijC ija
A (adjoint of A) ( )adj A
11 12 1 11 21 1
21 22 2 12 22 2
1 2 1 2
( )
T
n n
n n
n n nn n n nn
C C C C C C
C C C C C Cadj A
C C C C C C
= =
" "" "
# # # # # #" "
2.1.2 [ ]
ij n nA a = ija , 2n n
1. ( ) ( ) det( ) Aadj A adj A A A I= = 2. det( ) 0A 1 1 ( )
det( )A adj A
A =
Proof
-
| Page 6
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
9 A A = 1 0 12 1 0
1 1 1
10 3 2 1
5 6 2
1 0 3
A
=
11 12 131
21 22 23
31 32 33
b b b
A b b b
b b b
=
31 23b b+
-
| Page 7
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
fWhat you should learn -
-
- ERO
(Elementary Row Operation)
2.2.2 A n x n det (A) 0 AX = B x1 , x2 , , xn b1 , b2 , ,bn
X = 1
2
n
x
x
x
# , B =
1
2
n
b
b
b
#
x1 = 1det( )
det( )
A
A , x2 = 2
det( )
det( )
A
A, , xn = det( )
det( )nA
A
Ai i A B
11 2 3 9
2 3 3
x y
x y
+ = =
12
-
| Page 8
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
3 3
2 3 20
7 23
x y z
x y z
x y z
+ = + = + + =
-
| Page 9
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.1
1. 1 2 3
6 7 1
3 1 4
A
=
a. A b. A c. det(A) d. A-1
2.
4 0 0 1 0
3 3 3 1 0
1 2 4 2 3
9 4 6 2 3
2 2 4 2 3
A
=
1A
-
| Page 10
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
3-5 3. 3 2 13
3 2 5
x y
y
+ = =
4. 2 4 5
2 2 19
x y
x y
+ = + =
5. 2 3 8
4 2 4
3 2 9
x y z
x y z
x y z
+ = + + = + =
-
| Page 11
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.2 2.2.1 A A det( ) 0A = Proof 2.2.2 A det( ) det( )TA A= Proof 2.2.3 A 1. () A c
c det (A) 2. ( ) A
- det (A) 3. () A
() A () det(A)
-
| Page 12
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.2.4 A (Elementary Matrix) 1. E
nI k det( )E k=
2. E nI det( ) 1E =
3. E nI
det( ) 1E =
13 0 1 5
3 6 9
2 6 1
A
=
14 1 0 0 3
2 7 0 6
0 6 3 0
7 3 1 5
A
=
-
| Page 13
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.2 1. det( ) det( )TA A=
a. 2 31 4
A =
b. 1 2 3
6 7 1
3 1 4
A
=
2.
a. 3 17 4
0 5 1
0 0 2
b. 2 0 0 0
8 2 0 0
7 0 1 0
9 5 6 1
c. 1 0 0 0
0 1 0 0
0 0 5 0
0 0 0 1
d. 1 0 0 0
0 0 1 0
0 1 0 0
0 0 0 1
e. 1 0 0 0
0 1 0 9
0 0 1 0
0 0 0 1
-
| Page 14
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
3. a. 1 2 3
6 7 1
3 1 4
A
=
b. 2 1 3 11 0 1 10 2 1 0
0 1 2 3
A
=
4. 6a b c
d e f
g h i
=
a. d e f
g h i
a b c
b. 3 3 3
4 4 4
a b c
d e f
g h i
c. 3 3 3
4 4 4
a b c
d e f
g d h e i f
5. 2.2.4
-
| Page 15
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.3 2.3.1 A, B n x n k 1. det(kA) = kndet(A) 2. det(Ak) = [det(A)]k k
2.3.2 A, B C n x n r r C r A B det(C) = det(A) + det(B)
-
| Page 16
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.3.3 A n x n E n x n det(EA) = det(E) det(A)
2.3.4 A det(A) 0
-
| Page 17
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.3.5 A,B n x n det(AB) = det(A) det(B)
2.3.6 A 1 1det( )
det( )A
A =
2.3.7 A n x n
1. A 2. 3.
RR nA I=
4. A 5. AX B=
B 1n 6. AX B=
B 1n 7. det(A) 0
Note: 1. A = B det (A) = det (B) 2. det (A) = det (B) A = B 3. det (A+B) det (A) + det (B)
-
| Page 18
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
15 det(A) A = 1 1 1 2
1 2 2 1
4 3 0 1
3 0 2 1
16 det(A) A = 1 1 3 1
1 1 1 2
3 2 2 1
2 3 2 5
17 A = 3 2
2 1 2
0 3 2
x
x
B = 2
3 1
x x
x det(A) = 4 det(B)
-
| Page 19
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
18 A = 1 2 2 1 2 1 2 1, , ,3 4 3 0 4 3 4 3
A B C D = = = =
det(ABCD)
19 2 1 5 3 3 4 3 41 1 1 1 2 1 2 1
a b
c d
=
a bc d
-
| Page 20
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
A =x x
A =x x
( )n
n
A
A I
A I
= = =
x x 0x x 0
x 0
( )n
A I =x 0 () ( )
nA I =x 0
( ) (nontrivial solutions) (eigenvalue) A (eigenvector) A ( )
nA I =x 0
nA I
nA I ( )
nA I =x 0
det( ) 0n
A I =
20 1 2 1
1 2 2
3
4 2
x x x
x x x
+ = + =
-
| Page 21
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
2.3 1. det(kA)
a. 1 2 , 23 4
A k = =
b. 2 1 3
3 2 1 , 2
1 4 5
A k
= =
2. det(AB) 2 1 0 1 1 3
3 4 0 , 7 1 2
0 0 2 5 0 1
A B
= =
3. det(A)=0 2 8 1 4
3 2 5 1
1 10 6 5
4 6 4 3
A
=
4. k
a. 3 22 2
kA
k
=
b. 1 2 4
3 1 6
3 2
A
k
=
-
| Page 22
Created by Deaw JAIBUN, Mahidol Wittayanusorn School
5. i. ii. iii.
a. 1 2 1
1 2 2
2
2
x x x
x x x
+ = + =
b. 1 2 11 2 2
2 3
4 3
x x x
x x x
+ = + =