Question

求下列矩阵的逆矩阵，要过程解释。

Answer 1

(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