6 行列式の基本法則と効率的な ... - cis.twcu.ac.jp

1 (2002 ) 176 . , . , .1. 1 . 100010001 = 1 . a13a12a11a23a22a21a33a32a31 = a11a12a13a21a22a23a31a32a33 .3. c c . ca11a12a13ca21a22a23ca31a32a33 =c a11a12a13a21a22a23a31a32a33 .4. , . a11+b11a12a13a21+b21a22a23a31+b31a32a33 = a11a12a13a21a22a23a31a32a33 + b11a12a13b21a22a23b31a32a33 .. , , .5.. a11a21a31a12a22a32a13a23a33 = a11a12a13a21a22a23a31a32a33 . , .6. 0 . a11a12a11a21a22a21a31a32a31 = . a11+ca13a12a13a21+ca23a22a23a31+ca33a32a 33 = a11a12a13a21a22a23a31a32a33.

2 (2002 ) 18 6. 123235357 . 123235357 = 1032 153 17 = 10 02 1 13 1 2 = 1 1 1 2 = 1112 =1. 18.. - 1 0 . 3 1 .. 0 , . 0 . , - 0 , . 7. a111a111a = a+1 1 1a+1a121a = a+2 1 1a+2a1a+2 1a =(a+2) 1111a111a =(a+2) 1110a 1011a =(a+2) 11 10a 1000a 1 , 1 (a+2) 11 10a 1000a 1 =(a+2) a 100a 1 =(a+2)(a 1)2. 19.. 20. abbbabbba =(a+2b)(a b)2 . 21. 111abca2b2c2 =(a b)(b c)(c a).

3 A11x1+a12x2+a13x3=b1a21x1+a22x2+a23x3=b2 a31x1+a32x2+a33x3=b3 (2002 ) 19 D= a11a12a13a21a22a23a31a32a33 . 1 D1 . b1=a11x1+a12x2+a13x3 , D1= b1a12a13b2a22a23b3a32a33 = a11x1+a12x2+a13x3a12a13a21x1+a22x2+a23x3 a22a23a31x1+a32x2+a33x3a32a33 == a11x1a12a13a21x1a22a23a31x1a32a33 + a12x2a12a13a22x2a22a23a32x2a32a33 + a13x3a12a13a23x3a22a23a33x3a32a33 =x1 a11a12a13a21a22a23a31a32a33 +x2 0+x3 0=D x1. D1=Dx1. D2= a11b1a13a21b2a23a31b3a33 ,D3= a11a12b1a21a22b2a31a32b3 D2=Dx2,D3=Dx3. , D 0 x1=D1D,x2=D2D,x3=D3D. 22.. 23. D6=0 x1=D1D,x2=D2D,x3=D3D .D=0 D1=0,D2=0,D3=0 ,D1,D2,D3 0 . , D=0 .. 0 a11x1+a12x2+a13x3=0a21x1+a22x2+a23x3=0a3 1x1+a32x2+a33x3=0.

4 0 , 17 . ,x1=0,x2=0,x3=0 17 homogeneous. (2002 ) 20 , . D ,D=0 x1=x2=x3=0 . x1=x2=x3=0 D=0. , . , n x1=x2=..=xn=0 , 0 .. 24. a11x2+a12x+a13=0,a21x2+a22x+a23=0,a31x2+ a32x+a33=0 x a11a12a13a21a22a23a31a32a33 =0 .. , D= a11a12a13a21a22a23a31a32a33 , 1 j (j=1,2,3) A1j . 1 1 ,D0= a21a22a23a21a22a23a31a32a33 D0 1 j A1j . D0 1 D0=a21A11+a22A12+a23A13 . D0=0 ,a21A11+a22A12+a23A13=0 . D 2 1 0 . , , 0 . - , .. a11a12a13a21a22a23a31a32a33 b11b12b13b21b22b23b31b32b33 = (2002 ) 21 a11b11+a12b21+a13b31a11b12+a12b22+a13b32 a11b13+a12b23+a13b33a21b11+a22b21+a23b31 a21b12+a22b22+a23b32a21b13+a22b23+a23b33 a31b11+a32b21+a33b31a31b12+a32b22+a33b32 a31b13+a32b23+a33b33.

5 I j i j , . 25.. , . 8. D= a11a12a13a21a22a23a31a32a33 , i j (i=1,2, 3;j=1,2,3) Aij ,Aij j i A11A21A31A12A22A32A13A23A33 . a11a12a13a21a22a23a31a32a33 A11A21A31A12A22A32A13A23A33 = D000D000D =D3, A11A21A31A12A22A32A13A23A33 =D2. 26.. (2002 ) 227 , . 18 , . , & : 7 .. , , , . , . , . , , , . ,3 4 54.

6 6 6 1458 . 27.* .. , 19 , 20 , . (0,0), (a, c), (a+b, c+d), (b, d) abcd . , , (0,0) (a, c) (a+b, c+d) (b, d) (0,0) , . (0,0), (a, c), (a+b, c+d), (b, d) abcd =018C. F. Gauss, 1777 Descartes, 1596 . (Cartesian coordinates) . (2002 ) 23 . 28.. 29. (x1,y1), (x2,y2), (x3,y3) 12 111x1x2x3y1y2y3 . , (x1,y1) (x2,y2) (x3,y3) (x1,y1) , . (x1,y1),(x2,y2), (x3,y3) 111x1x2x3y1y2y3 =0 .. (x, y) (ax+cy, bx+dy) , abcd . , abcd 6=0 . & : 7 6.

7 30.* , . , .. , . , . ( ).