老师,这个行列式用递推法怎么算啊?
来源:学生作业帮 编辑:百度作业网作业帮 分类:数学作业 时间:2024/07/31 08:22:46
老师,这个行列式用递推法怎么算啊?
0 1 0 0…0 0
1 0 1 0…0 0
0 1 0 1…0 0
……
0 0 0 0…1 0
n行n列
0 1 0 0…0 0
1 0 1 0…0 0
0 1 0 1…0 0
……
0 0 0 0…1 0
n行n列
![老师,这个行列式用递推法怎么算啊?](/uploads/image/z/2780371-19-1.jpg?t=%E8%80%81%E5%B8%88%2C%E8%BF%99%E4%B8%AA%E8%A1%8C%E5%88%97%E5%BC%8F%E7%94%A8%E9%80%92%E6%8E%A8%E6%B3%95%E6%80%8E%E4%B9%88%E7%AE%97%E5%95%8A%3F)
按第1列展开
Dn = (-1)^(2+1) * 1 * (-1)^(1+1) * 1 * Dn-2 = - Dn-2
当 n=2k+1 (奇数) 时
Dn = -Dn-2 = (-1)^2 Dn-4 = ... = (-1)^k D1 = 0
当 n=2k (偶数) 时
Dn = -Dn-2 = (-1)^2 Dn-4 = ... = (-1)^(k-1) D2 = (-1)^k = (-1)^(n/2)
Dn = (-1)^(2+1) * 1 * (-1)^(1+1) * 1 * Dn-2 = - Dn-2
当 n=2k+1 (奇数) 时
Dn = -Dn-2 = (-1)^2 Dn-4 = ... = (-1)^k D1 = 0
当 n=2k (偶数) 时
Dn = -Dn-2 = (-1)^2 Dn-4 = ... = (-1)^(k-1) D2 = (-1)^k = (-1)^(n/2)