矩阵相乘的次数 | 相乘矩阵自乘的方法 | 幂乘的方法 | 快速Warshall转移法 |
1 |
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甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
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乙 |
1 |
1 |
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丙 |
1 |
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1 |
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1 |
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丁 |
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1 |
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1 |
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戊 |
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1 |
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1 |
1 |
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己 |
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1 |
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1 |
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庚 |
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
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1 |
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1 |
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癸 |
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1 |
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甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
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乙 |
1 |
1 |
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丙 |
1 |
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1 |
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1 |
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丁 |
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1 |
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1 |
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戊 |
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1 |
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1 |
1 |
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己 |
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1 |
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1 |
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庚 |
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
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1 |
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1 |
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癸 |
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1 |
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甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
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乙 |
1 |
1 |
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丙 |
1 |
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1 |
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1 |
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丁 |
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1 |
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1 |
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戊 |
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1 |
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1 |
1 |
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己 |
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1 |
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1 |
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庚 |
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
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1 |
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1 |
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癸 |
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1 |
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2 |
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甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
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乙 |
1 |
1 |
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丙 |
1 |
1 |
1 |
1 |
1 |
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丁 |
1 |
1 |
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1 |
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戊 |
1 |
1 |
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1 |
1 |
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己 |
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1 |
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1 |
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1 |
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庚 |
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
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1 |
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1 |
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1 |
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癸 |
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1 |
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甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
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乙 |
1 |
1 |
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丙 |
1 |
1 |
1 |
1 |
1 |
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丁 |
1 |
1 |
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1 |
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戊 |
1 |
1 |
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1 |
1 |
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己 |
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1 |
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1 |
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1 |
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庚 |
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
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1 |
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1 |
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1 |
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癸 |
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1 |
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甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
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乙 |
1 |
1 |
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丙 |
1 |
1 |
1 |
1 |
1 |
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丁 |
1 |
1 |
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1 |
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戊 |
1 |
1 |
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1 |
1 |
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己 |
1 |
1 |
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1 |
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1 |
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庚 |
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
1 |
1 |
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1 |
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1 |
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1 |
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癸 |
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1 |
|
3 |
|
甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
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乙 |
1 |
1 |
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丙 |
1 |
1 |
1 |
1 |
1 |
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丁 |
1 |
1 |
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1 |
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戊 |
1 |
1 |
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1 |
1 |
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己 |
1 |
1 |
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1 |
|
1 |
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庚 |
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
|
1 |
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1 |
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1 |
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1 |
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癸 |
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|
1 |
|
|
甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
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乙 |
1 |
1 |
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丙 |
1 |
1 |
1 |
1 |
1 |
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丁 |
1 |
1 |
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1 |
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戊 |
1 |
1 |
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1 |
1 |
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己 |
1 |
1 |
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1 |
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1 |
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庚 |
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
1 |
1 |
|
1 |
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1 |
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1 |
|
癸 |
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1 |
|
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甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
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|
|
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乙 |
1 |
1 |
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丙 |
1 |
1 |
1 |
1 |
1 |
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|
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|
丁 |
1 |
1 |
|
1 |
|
|
|
|
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戊 |
1 |
1 |
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1 |
1 |
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己 |
1 |
1 |
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1 |
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1 |
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庚 |
|
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1 |
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1 |
辛 |
1 |
1 |
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1 |
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壬 |
1 |
1 |
|
1 |
|
1 |
|
|
1 |
|
癸 |
|
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|
1 |
|
4 |
|
甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
|
|
|
|
|
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|
乙 |
1 |
1 |
|
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|
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丙 |
1 |
1 |
1 |
1 |
1 |
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|
丁 |
1 |
1 |
|
1 |
|
|
|
|
|
|
戊 |
1 |
1 |
|
1 |
1 |
|
|
|
|
|
己 |
1 |
1 |
|
1 |
|
1 |
|
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|
庚 |
|
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|
1 |
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1 |
辛 |
1 |
1 |
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|
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|
1 |
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|
壬 |
1 |
1 |
|
1 |
|
1 |
|
|
1 |
|
癸 |
|
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|
|
|
|
|
|
1 |
|
|
甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
|
|
|
|
|
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|
乙 |
1 |
1 |
|
|
|
|
|
|
|
|
丙 |
1 |
1 |
1 |
1 |
1 |
|
|
|
|
|
丁 |
1 |
1 |
|
1 |
|
|
|
|
|
|
戊 |
1 |
1 |
|
1 |
1 |
|
|
|
|
|
己 |
1 |
1 |
|
1 |
|
1 |
|
|
|
|
庚 |
|
|
|
|
|
|
1 |
|
|
1 |
辛 |
1 |
1 |
|
|
|
|
|
1 |
|
|
壬 |
1 |
1 |
|
1 |
|
1 |
|
|
1 |
|
癸 |
|
|
|
|
|
|
|
|
|
1 |
| |
5 |
|
甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
|
|
|
|
|
|
|
乙 |
1 |
1 |
|
|
|
|
|
|
|
|
丙 |
1 |
1 |
1 |
1 |
1 |
|
|
|
|
|
丁 |
1 |
1 |
|
1 |
|
|
|
|
|
|
戊 |
1 |
1 |
|
1 |
1 |
|
|
|
|
|
己 |
1 |
1 |
|
1 |
|
1 |
|
|
|
|
庚 |
|
|
|
|
|
|
1 |
|
|
1 |
辛 |
1 |
1 |
|
|
|
|
|
1 |
|
|
壬 |
1 |
1 |
|
1 |
|
1 |
|
|
1 |
|
癸 |
|
|
|
|
|
|
|
|
|
1 |
| | |
矩阵相乘的次数 | 相乘矩阵自乘的方法 | 幂乘的方法 | 快速Warshall转移法 |
1 |
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、丙、戊、 |
丁 |
乙、丁、 |
戊 |
乙、丁、戊、 |
己 |
丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
己、壬、 |
癸 |
癸、 |
|
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、丙、戊、 |
丁 |
乙、丁、 |
戊 |
乙、丁、戊、 |
己 |
丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
己、壬、 |
癸 |
癸、 |
|
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、丙、戊、 |
丁 |
乙、丁、 |
戊 |
乙、丁、戊、 |
己 |
丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
己、壬、 |
癸 |
癸、 |
|
2 |
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
丁、己、壬、 |
癸 |
癸、 |
|
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
丁、己、壬、 |
癸 |
癸、 |
|
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
甲、乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
甲、乙、丁、己、壬、 |
癸 |
癸、 |
|
3 |
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
甲、乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
乙、丁、己、壬、 |
癸 |
癸、 |
|
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
甲、乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
甲、乙、丁、己、壬、 |
癸 |
癸、 |
|
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
甲、乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
甲、乙、丁、己、壬、 |
癸 |
癸、 |
|
4 |
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
甲、乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
甲、乙、丁、己、壬、 |
癸 |
癸、 |
|
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
甲、乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
甲、乙、丁、己、壬、 |
癸 |
癸、 |
| |
5 |
甲 |
甲、乙、 |
乙 |
甲、乙、 |
丙 |
甲、乙、丙、丁、戊、 |
丁 |
甲、乙、丁、 |
戊 |
甲、乙、丁、戊、 |
己 |
甲、乙、丁、己、 |
庚 |
庚、癸、 |
辛 |
甲、乙、辛、 |
壬 |
甲、乙、丁、己、壬、 |
癸 |
癸、 |
| | |
矩阵相乘的次数 | 相乘矩阵自乘的方法 | 逐次平方法 | 快速转移法,Warshall快速转移 |
1 |
|
甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
|
|
|
|
|
|
|
乙 |
1 |
1 |
|
|
|
|
|
|
|
|
丙 |
1 |
|
1 |
|
1 |
|
|
|
|
|
丁 |
|
1 |
|
1 |
|
|
|
|
|
|
戊 |
|
1 |
|
1 |
1 |
|
|
|
|
|
己 |
|
|
|
1 |
|
1 |
|
|
|
|
庚 |
|
|
|
|
|
|
1 |
|
|
1 |
辛 |
1 |
1 |
|
|
|
|
|
1 |
|
|
壬 |
|
|
|
|
|
1 |
|
|
1 |
|
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
|
甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
|
|
|
|
|
|
|
乙 |
1 |
1 |
|
|
|
|
|
|
|
|
丙 |
1 |
|
1 |
|
1 |
|
|
|
|
|
丁 |
|
1 |
|
1 |
|
|
|
|
|
|
戊 |
|
1 |
|
1 |
1 |
|
|
|
|
|
己 |
|
|
|
1 |
|
1 |
|
|
|
|
庚 |
|
|
|
|
|
|
1 |
|
|
1 |
辛 |
1 |
1 |
|
|
|
|
|
1 |
|
|
壬 |
|
|
|
|
|
1 |
|
|
1 |
|
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
|
甲 | 乙 | 丙 | 丁 | 戊 | 己 | 庚 | 辛 | 壬 | 癸 |
甲 |
1 |
1 |
|
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|
|
|
|
|
|
乙 |
1 |
1 |
|
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|
|
|
|
|
|
丙 |
1 |
|
1 |
|
1 |
|
|
|
|
|
丁 |
|
1 |
|
1 |
|
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|
|
|
|
戊 |
|
1 |
|
1 |
1 |
|
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|
|
己 |
|
|
|
1 |
|
1 |
|
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|
庚 |
|
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|
|
|
|
1 |
|
|
1 |
辛 |
1 |
1 |
|
|
|
|
|
1 |
|
|
壬 |
|
|
|
|
|
1 |
|
|
1 |
|
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
2 |
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 |
1 | 1 |
1 | |
|
|
|
|
丁 |
1 |
1 | |
1 | |
|
|
|
|
|
戊 |
1 |
1 | |
1 | 1 | |
|
|
|
|
己 |
|
1 |
|
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
|
|
|
1 |
|
1 | |
|
1 | |
癸 |
|
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|
|
|
|
|
|
|
1 |
|
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 |
1 | 1 |
1 | |
|
|
|
|
丁 |
1 |
1 | |
1 | |
|
|
|
|
|
戊 |
1 |
1 | |
1 | 1 | |
|
|
|
|
己 |
|
1 |
|
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
|
|
|
1 |
|
1 | |
|
1 | |
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 |
1 | 1 |
1 | |
|
|
|
|
丁 |
1 |
1 | |
1 | |
|
|
|
|
|
戊 |
1 |
1 | |
1 | 1 | |
|
|
|
|
己 |
1 |
1 |
|
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
1 |
1 |
|
1 |
|
1 | |
|
1 | |
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
3 |
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 | 1 | 1 | 1 | |
|
|
|
|
丁 |
1 | 1 | |
1 | |
|
|
|
|
|
戊 |
1 | 1 | |
1 | 1 | |
|
|
|
|
己 |
1 |
1 | |
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
|
1 |
|
1 | |
1 | |
|
1 | |
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 | 1 | 1 | 1 | |
|
|
|
|
丁 |
1 | 1 | |
1 | |
|
|
|
|
|
戊 |
1 | 1 | |
1 | 1 | |
|
|
|
|
己 |
1 |
1 | |
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
1 |
1 |
|
1 | |
1 | |
|
1 | |
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 | 1 | 1 | 1 | |
|
|
|
|
丁 |
1 | 1 | |
1 | |
|
|
|
|
|
戊 |
1 | 1 | |
1 | 1 | |
|
|
|
|
己 |
1 | 1 | |
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
1 | 1 | |
1 | |
1 | |
|
1 | |
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
4 |
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 | 1 | 1 | 1 | |
|
|
|
|
丁 |
1 | 1 | |
1 | |
|
|
|
|
|
戊 |
1 | 1 | |
1 | 1 | |
|
|
|
|
己 |
1 | 1 | |
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
1 |
1 | |
1 | |
1 | |
|
1 | |
癸 |
|
|
|
|
|
|
|
|
|
1 |
|
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 | 1 | 1 | 1 | |
|
|
|
|
丁 |
1 | 1 | |
1 | |
|
|
|
|
|
戊 |
1 | 1 | |
1 | 1 | |
|
|
|
|
己 |
1 | 1 | |
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
1 | 1 | |
1 | |
1 | |
|
1 | |
癸 |
|
|
|
|
|
|
|
|
|
1 |
| |
5 |
|
甲 |
乙 |
丙 |
丁 |
戊 |
己 |
庚 |
辛 |
壬 |
癸 |
甲 |
1 | 1 | |
|
|
|
|
|
|
|
乙 |
1 | 1 | |
|
|
|
|
|
|
|
丙 |
1 | 1 | 1 | 1 | 1 | |
|
|
|
|
丁 |
1 | 1 | |
1 | |
|
|
|
|
|
戊 |
1 | 1 | |
1 | 1 | |
|
|
|
|
己 |
1 | 1 | |
1 | |
1 | |
|
|
|
庚 |
|
|
|
|
|
|
1 | |
|
1 |
辛 |
1 | 1 | |
|
|
|
|
1 | |
|
壬 |
1 | 1 | |
1 | |
1 | |
|
1 | |
癸 |
|
|
|
|
|
|
|
|
|
1 |
| | |