1) This document is a series of mathematical equations and explanations regarding linear algebra concepts such as systems of linear equations, matrices, determinants, and Cramer's rule. 2) Several examples are provided to demonstrate calculating the determinant of matrices using Laplace expansions and cofactor expressions. 3) Cramer's rule for solving systems of linear equations using determinants and the inverse of a matrix is explained. References to other sections and pages for further information are provided.

1. II
shito@seinan-gu.ac.jp
2020 5 8
•
•
•
•
1
(1)
1
⇐⇒ 1
⇑
(a) ⇐⇒
(b) P.5
10 4
5 2
x1
x2
=
d1
d2
1

2. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
i. d1 = 2d2 2
d1 = 12 d2 = 6 10x1 + 4x2 = 12
ii. d1 = 2d2 2
10x1 + 4x2 = 12 5x1 + 2x2 = 0
• ⇐⇒
II 2

3. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
• ⇐⇒
1
⇓
(2) (rank)
: 1
⇐⇒
(1) pp.105–110
(2) 5.1
2 (determinant)
1
(determinant)
(1)
A |A|
A =
a b
c d
|A| =
B =
4 8
2 5
|B| =
C =
4 8
2 4
|C| =
D =
4 8
4 8
|D| =
1
II 3

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(2)
II 4

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(3) 3
A =



a11 a12 a13
a21 a22 a23
a31 a32 a33



(a)
3



a11 a12 a13
a21 a22 a23
a31 a32 a33



II 5

6. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
(b) (Laplace expansion)
|A| =
a11 a12 a13
a21 a22 a23
a31 a32 a33
Cofactor minor
|Cij| ≡
|C11| =
|C12| =
|C13| =
|A| = a11|M11| − a12|M12| + a13|M13|
= a11|C11| + a12|C12| + a13|C13|
=
3
j=1
a1j|C1j|
1 or
II 6

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:
5 6 1
2 3 0
7 −3 0
(4) 3
II 7

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(5) n
3
a11 a12 a13 a14
a21 a22 a23 a24
a31 a32 a33 a34
a41 a42 a43 a44
=
(6) n
2 3 n
1
(1) pp.110–116
(2) 5.2 1–4
II 8

9. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
3
(1) I
|A| = |A |
=⇒
=⇒
(2) III
k k
=⇒ k
k
=⇒ k
28 63 21
32 53 17
4 23 9
=
II 9

10. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
(3) IV
7 × 4
1 9 3
8 53 17
1 23 9
=
(4) V
2
IV
=⇒
(5)
A n × n Ax = d
|A| = 0 ⇐⇒
⇐⇒
⇐⇒
⇐⇒
II 10

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(1) pp.117–122
(2) 5.3 1–6
4
Ax = d
↔ ¯x = A−1
d A
A−1
(1)
|A| =
a11 a12 a13
a21 a22 a23
a31 a32 a33
1 2
3
j=1
a1j|C2j| = a11|C21| + a12|C22| + a13|C23|
=
VI
0
a11 a12 a13
a11 a12 a13
a31 a32 a33
1



n
j=1
aij|Ci j| = 0 (i = i ) i i
n
i=1
aij|Cij | = 0 (j = j ) j j
II 11

12. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
(2)
(a)
A
n×n
=






a11 a12 . . . a1n
a21 a22 . . . a2n
. . . . . . . . . . . . . . .
an1 an2 . . . ann






=⇒
(b) C = (|Cij|)
C
n×n
=






|C11| |C12| . . . |C1n|
|C21| |C22| . . . |C2n|
. . . . . . . . . . . . . . . . . .
|Cn1| |Cn2| . . . |Cnn|






(c) (adj A) the adjoint of A
C
n×n
≡ adj A ≡
(d) AC
AC
n×n
=
II 12

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(e)
AC = |A|I
A−1
=
II 13

14. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
: A =
a b
c d
(1) pp.123–127
(2) 5.4 1–3
1
Suppose that we expand a fourth-order determinant by its third column and the cofactors
of the second-column elements. How would you write the resulting sum of products in
notation? What will be the sum of products in notation if we expand it by the second
row and the cofactors of the fourth-row elements?
II 14

15. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
5 (Cramer’s rule)
(1)
A
n×n
x = d
↔






¯x1
¯x2
...
¯xn






=
1
|A|






|C11| |C21| . . . |Cn1|
|C12| |C22| . . . |Cn2|
. . . . . . . . . . . . . . . . . .
|C1n| |C2n| . . . |Cnn|












d1
d2
...
dn






=
1
|A|






n
i=1 di|Ci1|
n
i=1 di|Ci2|
...
n
i=1 di|Cin|






1 ¯x1 =
n
i=1 di|Ci1|
|A|
|A| 1
|A| 1 di |A1| |A1| =
¯x1 =
¯xj =
II 15

16. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27
: Cramer’s rule
5x1 + 3x2 = 30
6x1 − 2x2 = 8
II 16

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(1) pp.127–133 5.6
(2) 5.5 1–3
(3) 5.7 1
(4) p.56 (3.12) 2
6
=⇒
• =
=⇒
• =
=⇒
(1) pp.144–145
II 17