Prove that the following determinant equals zero without expanding the determinant
︱a-b b-c c-a︱
︱b-c c-a a-b︱
︱c-a a-b b-c︱
Matrix Question
2013-04-19 1:40 am
回答 (2)
2013-04-19 2:36 am
✔ 最佳答案
Prove that the following determinant equalszero without expanding the determinant ｜ａ－ｂ ｂ－ｃ ｃ－ａ｜
｜ｂ－ｃ ｃ－ａ ａ－ｂ｜
｜ｃ－ａ ａ－ｂ ｂ－ｃ｜
Sol｜ａ－ｂ ｂ－ｃ ｃ－ａ｜＊１──────
｜ｂ－ｃ ｃ－ａ ａ－ｂ｜＜───────｜
｜ｃ－ａ ａ－ｂ ｂ－ｃ｜
｜ ａ－ｂ ｂ－ｃ ｃ－ａ｜
＝｜ａ－ｂ＋ｂ－ｃ ｂ－ｃ＋ｃ－ａ ｃ－ａ＋ａ－ｂ｜
｜ ｃ－ａ ａ－ｂ ｂ－ｃ｜
｜ａ－ｂ ｂ－ｃ ｃ－ａ｜
＝｜ａ－ｃ ｂ－ａ ｃ－ｂ｜＊１──────
｜ｃ－ａ ａ－ｂ ｂ－ｃ｜＜───────｜
｜ ａ－ｂ ｂ－ｃ ｃ－ａ｜
＝｜ ａ－ｃ ｂ－ａ ｃ－ｂ｜
｜ａ－ｃ＋ｃ－ａ ｂ－ａ＋ａ－ｂ ｃ－ｂ＋ｂ－ｃ｜
｜ａ－ｂ ｂ－ｃ ｃ－ａ｜
＝｜ａ－ｃ ｂ－ａ ｃ－ｂ｜
｜ ０ ０ ０｜
=0
2013-04-19 5:03 am
| a - b b - c c - a |
| b - c c - a a - b |
| c - a a - b b - c |
= | a - b a - c c - a | C1 + C2 to C2
| b - c b - a a - b |
| c - a c - b b - c |
= | a - b a - c 0 | C2 + C3 to C3
| b - c b - a 0 |
| c - a c - b 0 |
= 0.
| b - c c - a a - b |
| c - a a - b b - c |
= | a - b a - c c - a | C1 + C2 to C2
| b - c b - a a - b |
| c - a c - b b - c |
= | a - b a - c 0 | C2 + C3 to C3
| b - c b - a 0 |
| c - a c - b 0 |
= 0.
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