n阶段矩阵计算设A、B均为n阶矩阵,且丨A丨=3,丨B丨=-2,A*B*分别为AB的伴随矩阵,则丨2A^(-1)B*+A*B^(-1)丨=?丨2A^(-1)B*+A*B^(-1)丨=?=丨(2/3)A*B*-(1/2)A*B*丨=丨(1/6)A*B*丨=(1/6)^n丨A*B*丨=(1/6)^n丨A*丨
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![n阶段矩阵计算设A、B均为n阶矩阵,且丨A丨=3,丨B丨=-2,A*B*分别为AB的伴随矩阵,则丨2A^(-1)B*+A*B^(-1)丨=?丨2A^(-1)B*+A*B^(-1)丨=?=丨(2/3)A*B*-(1/2)A*B*丨=丨(1/6)A*B*丨=(1/6)^n丨A*B*丨=(1/6)^n丨A*丨](/uploads/image/z/9915820-52-0.jpg?t=n%E9%98%B6%E6%AE%B5%E7%9F%A9%E9%98%B5%E8%AE%A1%E7%AE%97%E8%AE%BEA%E3%80%81B%E5%9D%87%E4%B8%BAn%E9%98%B6%E7%9F%A9%E9%98%B5%2C%E4%B8%94%E4%B8%A8A%E4%B8%A8%3D3%2C%E4%B8%A8B%E4%B8%A8%3D-2%2CA%2AB%2A%E5%88%86%E5%88%AB%E4%B8%BAAB%E7%9A%84%E4%BC%B4%E9%9A%8F%E7%9F%A9%E9%98%B5%2C%E5%88%99%E4%B8%A82A%5E%28-1%29B%2A%2BA%2AB%5E%28-1%29%E4%B8%A8%3D%3F%E4%B8%A82A%5E%28-1%29B%2A%2BA%2AB%5E%28-1%29%E4%B8%A8%3D%3F%3D%E4%B8%A8%EF%BC%882%2F3%EF%BC%89A%2AB%2A-%EF%BC%881%2F2%EF%BC%89A%2AB%2A%E4%B8%A8%3D%E4%B8%A8%EF%BC%881%2F6%EF%BC%89A%2AB%2A%E4%B8%A8%3D%EF%BC%881%2F6%EF%BC%89%5En%E4%B8%A8A%2AB%2A%E4%B8%A8%3D%EF%BC%881%2F6%EF%BC%89%5En%E4%B8%A8A%2A%E4%B8%A8)
n阶段矩阵计算设A、B均为n阶矩阵,且丨A丨=3,丨B丨=-2,A*B*分别为AB的伴随矩阵,则丨2A^(-1)B*+A*B^(-1)丨=?丨2A^(-1)B*+A*B^(-1)丨=?=丨(2/3)A*B*-(1/2)A*B*丨=丨(1/6)A*B*丨=(1/6)^n丨A*B*丨=(1/6)^n丨A*丨
n阶段矩阵计算
设A、B均为n阶矩阵,且丨A丨=3,丨B丨=-2,A*B*分别为AB的伴随矩阵,则丨2A^(-1)B*+A*B^(-1)丨=?
丨2A^(-1)B*+A*B^(-1)丨=?
=丨(2/3)A*B*-(1/2)A*B*丨
=丨(1/6)A*B*丨
=(1/6)^n丨A*B*丨
=(1/6)^n丨A*丨丨B*丨
丨A*丨=3^(n-1)丨B*丨=(-2)^(n-1)带入即可
请问我的解答方法那里错了,应该如何修改
n阶段矩阵计算设A、B均为n阶矩阵,且丨A丨=3,丨B丨=-2,A*B*分别为AB的伴随矩阵,则丨2A^(-1)B*+A*B^(-1)丨=?丨2A^(-1)B*+A*B^(-1)丨=?=丨(2/3)A*B*-(1/2)A*B*丨=丨(1/6)A*B*丨=(1/6)^n丨A*B*丨=(1/6)^n丨A*丨
你做的对!
也可用 A* = |A|A^-1
丨2A^(-1)B*+A*B^(-1)丨
= | 2|B|A^-1B^-1+|A|A^-1B^-1丨
= | - A^-1B^-1 |
= (-1)^n (-1/6).
A[2A^(-1)B*+A*B^(-1)]B
= 2AA^(-1)B*B+AA*B^(-1)B
= 2|B|E + |A|E
= -E.
等式两边取行列式得
|A||2A^(-1)B*+A*B^(-1)||B| = |-E|.
即有 -6|2A^(-1)B*+A*B^(-1)| = (-1)^n
故 |2A^(-1)B*+A*B^(-1)| = (-1/6)(-1)^n = (1/6)(-1)^(n-1).