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# 経済数学II 「第５章 線型モデルと行列代数 II」

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### 経済数学II 「第５章 線型モデルと行列代数 II」

1. 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. 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. 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
4. 4. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27 (2) II 4
5. 5. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27 (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. 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
7. 7. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27 : 5 6 1 2 3 0 7 −3 0 (4) 3 II 7
8. 8. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27 (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. 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. 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
11. 11. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27 (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. 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
13. 13. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27 (e) AC = |A|I A−1 = II 13
14. 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. 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. 16. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27 : Cramer’s rule 5x1 + 3x2 = 30 6x1 − 2x2 = 8 II 16
17. 17. www.seinan-gu.ac.jp/˜shito 2020 5 8 22:27 (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
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Apr. 27, 2021
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