Question

一道线性代数题

Dn=1+a1 1 ……1
1 1+a2……1 的行列式
……
1 1 …… 1+an

Answer 1

按最后一列分拆成两个行列式
Dn = a1a2...an-1 + anDn-1
= a1a2...an-1 + an(a1a2....an-2+an-1Dn-2) ---迭代
= a1a2...an-1 + a1a2....an-2an + an-1anDn-2
= ....
= a1a2...an-1 + a1a2....an-2an + ... + a2...an-1anD1
= a1a2...an-1 + a1a2....an-2an + ... + a2...an-1an(1+a1)
= a1a2...an-1 + a1a2....an-2an + ... + a2...an-1an + a1a2...an
= a1a2....an (1 + 1/a1 +1/a2+ ... + 1/an)