求下列矩阵的逆矩阵,
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求下列矩阵的逆矩阵,
![](http://img.wesiedu.com/upload/6/88/688a6e8344bf6e9a7e731d954a339495.jpg)
![](http://img.wesiedu.com/upload/6/88/688a6e8344bf6e9a7e731d954a339495.jpg)
![求下列矩阵的逆矩阵,](/uploads/image/z/17442754-34-4.jpg?t=%E6%B1%82%E4%B8%8B%E5%88%97%E7%9F%A9%E9%98%B5%E7%9A%84%E9%80%86%E7%9F%A9%E9%98%B5%2C%26nbsp%3B)
(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
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