Question

用矩阵的初等变换计算下列矩阵的逆矩阵 第一行是2 1 -1，第二行是0 2 1，第三行是5 2 -3

Answer 1

解: (A,E) =
2 1 -1 1 0 0
0 2 1 0 1 0
5 2 -3 0 0 1

r3-2r1
2 1 -1 1 0 0
0 2 1 0 1 0
1 0 -1 -2 0 1

r1-2r3
0 1 1 5 0 -2
0 2 1 0 1 0
1 0 -1 -2 0 1

r2-2r1
0 1 1 5 0 -2
0 0 -1 -10 1 4
1 0 -1 -2 0 1

r1+r2,r3-r2
0 1 0 -5 1 2
0 0 -1 -10 1 4
1 0 0 8 -1 -3

r2*(-1), 交换行
1 0 0 8 -1 -3
0 1 0 -5 1 2
0 0 1 10 -1 -4

所以 A^-1 =
8 -1 -3
-5 1 2
10 -1 -4

Answer 2

2 1 -1 | 1 0 0
0 2 1 | 0 1 0
5 2 -3 | 0 0 1

2 1 -1 | 1 0 0
0 2 1 | 0 1 0
5 0 -4 | 0 -1 1

2 0 -2/3 | 1 -1/2 0
0 1 1/2 | 0 1/2 0
5 0 -4 | 0 -1 1

1 0 -1/3 | 1/2 -1/4 0
0 1 1/2 | 0 1/2 0
1 0 -4/5 | 0 -1/5 1/5

1 0 -1/3 | 1/2 -1/4 0
0 1 1/2 | 0 1/2 0
0 0 -7/15 | -1/2 1/20 1/5

1 0 -1/3 | 1/2 -1/4 0
0 1 1/2 | 0 1/2 0
0 0 1 | 15/14 3/28 3/7

1 0 0 | 6/7 -3/14 1/7
0 1 0 | -15/7 1/14 -3/14
0 0 1 | 15/14 3/28 3/7

这个数值太难算了吧