利用行列式的性质证明| 1 1 1 || t t^2 t^3 |=t^4(t-1)^3(t+1) [这里的数字都是次方)
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利用行列式的性质证明
| 1 1 1 |
| t t^2 t^3 |=t^4(t-1)^3(t+1) [这里的数字都是次方)
| t^2 t^4 t^6 |
| 1 1 1 |
| t t^2 t^3 |=t^4(t-1)^3(t+1) [这里的数字都是次方)
| t^2 t^4 t^6 |
![利用行列式的性质证明| 1 1 1 || t t^2 t^3 |=t^4(t-1)^3(t+1) [这里的数字都是次方)](/uploads/image/z/15848537-41-7.jpg?t=%E5%88%A9%E7%94%A8%E8%A1%8C%E5%88%97%E5%BC%8F%E7%9A%84%E6%80%A7%E8%B4%A8%E8%AF%81%E6%98%8E%7C+1+1+1+%7C%7C+t+t%5E2+t%5E3+%7C%3Dt%5E4%28t-1%29%5E3%28t%2B1%29+%5B%E8%BF%99%E9%87%8C%E7%9A%84%E6%95%B0%E5%AD%97%E9%83%BD%E6%98%AF%E6%AC%A1%E6%96%B9%29)
证明:
| 1 1 1 |
| t t^2 t^3 |
| t^2 t^4 t^6 |
=1*t^2*t^6+1*t^3*t^2+1*t*t^4-1*t^2*t^2-1*t*t^6-1*t^3*t^4
=t^8+2t^5-t^4-2t^7
=t^4(t^4+2t-1-2t^3)
=t^4[(t^4-1)-2(t^3-t)]
=t^4[(t^2-1)(t^2+1)-2t(t^2-1)]
=t^4[(t^2-1)(t^2-2t+1)]
=t^4[(t-1)(t+1)(t-1)(t-1)]
=t^4(t-1)^3(t+1)
证毕.
| 1 1 1 |
| t t^2 t^3 |
| t^2 t^4 t^6 |
=1*t^2*t^6+1*t^3*t^2+1*t*t^4-1*t^2*t^2-1*t*t^6-1*t^3*t^4
=t^8+2t^5-t^4-2t^7
=t^4(t^4+2t-1-2t^3)
=t^4[(t^4-1)-2(t^3-t)]
=t^4[(t^2-1)(t^2+1)-2t(t^2-1)]
=t^4[(t^2-1)(t^2-2t+1)]
=t^4[(t-1)(t+1)(t-1)(t-1)]
=t^4(t-1)^3(t+1)
证毕.
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