Determinants

D= (-bc) (-ca) (-ab) + (ca + a²)(ab + b²)(bc + c²) + (ab + a²)(bc + b²)(ca + c²)
- (-bc) (ab + b²) (ca + c²) - (ca + a²)(bc + b²)(- ab) - (ab + a²)(- ca)(bc + c²)
= - (abc)² + abc (c + a)(a + b)(b + c) + abc (b + a)(c + b)(a + c)
+ (bc)² (a + b)(a + c) + (ab)² (c + a)(c + b) + (ca)² (b + a)(b + c)
= 2 abc (a + b)(b + c)(c + a)
+ (bc)² (a + b)(a + c) + (ab)² (c + a)(c + b) + (ca)² (b + a)(b + c)
- (abc)²
= 2 abc (b + c) (a² + ab + bc + ca)
+ (bc)² (a² + ab + bc + ca) + (ab)² (c² + ab + bc + ca) + (ca)² (b² + ab + bc + ca)
- (abc)²
= (ab + bc + ca) [ 2 abc (b + c) + (bc)² + (ab)² + (ca)² ]
+ 2 a³bc (b + c) + 3(abc)²
- (abc)²
= (ab + bc + ca) [ 2 abc (b + c) + (bc)² + (ab)² + (ca)² ] + 2a³bc (b + c) + 2(abc)²
= (ab + bc + ca) [ 2 abc (b + c) + (bc)² + (ab)² + (ca)² ] + 2a²bc (ab + ac + bc)
= (ab + bc + ca) [ (bc)² + (ab)² + (ca)² + 2 abc (b + c) + 2a²bc ]
= (ab + bc + ca) [ (bc)² + (ab)² + (ca)² + 2 abc (b + c + a) ]
= (ab + bc + ca) [ (bc)² + (ab)² + (ca)² + 2 (ab bc + bc ca + ca ab) ]
= (ab + bc + ca) (ab + bc + ca)²
= (ab + bc + ca)³

2011-09-15 13:38:08 補充：
Method 2 :

| - bc ..... ca + a² .....ab + a² |
| bc + b² .... - ca ..... ab + b² |
| bc + c² .... ca + c² .....- ab ..|

=

| - bc / a ..... c + a .....b + a |
| c + b .... - ca / b ..... a + b | * a b c
| b + c .... a + c .....- ab / c |

=

2011-09-15 13:38:51 補充：
| - bc ..... bc + ab .....bc + ca |
| ca + ab .... - ca ..... ac + bc | * a b c / (a b c)
| ab + ca .... ab + bc .....- ab |

=

| - bc ........ ab ..................bc + ca |
| ca + ab ... ab ................ ac + bc |
| ab + ca ....2ab + bc + ca .....- ab |

=

2011-09-15 13:39:14 補充：
| - bc - ca - ab ........ 0 ........................0........|
|.......... 0 ....... - ab - bc - ca ......ac + bc + ab|
| ab + ca ........2ab + bc + ca ..........- ab........|

=

(- bc - ca - ab) (- ab - bc - ca) (- ab)
- (- bc - ca - ab) (ac + bc + ab) (2ab + bc + ca)

2011-09-15 13:39:26 補充：
=

(ab + bc + ca)² (- ab)
+ (ab + bc + ca)² (2ab + bc + ca)

= (ab + bc + ca)³

2011-09-15 14:11:23 補充：
Method 3 :
By method 2 :

| - bc ..... bc + ab .....bc + ca |
| ca + ab .... - ca ..... ac + bc | * a b c / (a b c)
| ab + ca .... ab + bc .....- ab |

=

2011-09-15 14:11:34 補充：
| - bc ........ bc + ab ......ab + bc + ca |
| ca + ab .... - ca ..........ab + bc + ca |
| ab + ca ....ab + bc .....ab + bc + ca |

=

| - bc ........ bc + ab ......1 |
| ca + ab .... - ca ..........1 | * (ab + bc + ca)
| ab + ca ....ab + bc .....1 |

=

2011-09-15 14:11:40 補充：
| - bc .................... bc + ab .........1 |
| ca + ab + bc .... - ca - bc - ab ...0 | * (ab + bc + ca)
| ab + ca + bc ...............0 ...........0 |

= - 1 (- ca - bc - ab ) ( ab + ca + bc ) * (ab + bc + ca)

= (ab + bc + ca)³