求下列矩阵的逆矩阵,
问题描述：
求下列矩阵的逆矩阵,
答
(1)
对 (A,E) 用初等行变换化为 (E,A^-1)
(A,E)=
0 0 ...0 0 1 1 0 ...0 0 0
0 0 ...0 1 1 0 1 ...0 0 0
0 0 ...1 1 1 0 0 ...0 0 0
......
0 1 ...1 1 1 0 0 ...0 1 0
1 1 ...1 1 1 0 0 ...0 0 1
从最后一行开始,每行减上一行
0 0 ...0 0 1 1 0 ...0 0 0
0 0 ...0 1 0 -1 1 ...0 0 0
0 0 ...1 0 0 0 -1 ...0 0 0
......
0 1 ...0 0 0 0 0 ...-1 1 0
1 0 ...0 0 0 0 0 ...0 -1 1
交换行(上下倒置)
1 0 ...0 0 0 0 0 ...0 -1 1
0 1 ...0 0 0 0 0 ...-1 1 0
0 0 ...1 0 0 0 -1 ...0 0 0
0 0 ...0 1 0 -1 1 ...0 0 0
0 0 ...0 0 1 1 0 ...0 0 0
所以 A^-1=
0 0 ...0 -1 1
0 0 ...-1 1 0
0 -1 ...0 0 0
-1 1 ...0 0 0
1 0 ...0 0 0
(2) A^-1=
Ak^-1
Ak-1^-1
...
A1^-1
两个矩阵相乘等于E