Question

帮忙解一下这个行列式

Answer 1

将所有行加到第n行
1-a1 a2 0 0 ... 0 0
-1 1-a2 a3 0 ... 0 0
0 -1 1-a3 a4... 0 0
... ...
-a1 0 0 0 ... 0 1
按第n行展开
Dn = (-1)^(n+1)(-a1a2a3...an) + Dn-1
= (-1)^na1a2a3...an + Dn-1
--迭代
= (-1)^na1a2...an + (-1)^(n-1)a1a2...an-1 + Dn-2
= ......
= (-1)^na1a2...an + (-1)^(n-1)a1a2...an-1 + ... + (-1)^2a1a2+D1
= (-1)^na1a2...an + (-1)^(n-1)a1a2...an-1 + ... + (-1)^2a1a2 - a1 + 1
--写好看点
= 1-a1+a1a2-a1a2a3+...+(-1)^na1a2...an.