求 逆矩阵 用初等变换判定下列矩阵是否可逆,如可逆,求其逆矩阵.

题目

求 逆矩阵 用初等变换判定下列矩阵是否可逆,如可逆,求其逆矩阵.
｜ 3 2 1 ｜
｜ 3 1 5 ｜
｜ 3 2 3 ｜

答案

(A,E) =
3 2 1 1 0 0
3 1 5 0 1 0
3 2 3 0 0 1
r2-r1,r3-r1
3 2 1 1 0 0
0 -1 4 -1 1 0
0 0 2 -1 0 1
r1*(1/3),r2*(-1),r3*(1/2)
1 2/3 1/3 1/3 0 0
0 1 -4 1 -1 0
0 0 1 -1/2 0 1/2
r1-(1/3)r3,r2+4r3
1 2/3 0 1/2 0 -1/6
0 1 0 -1 -1 2
0 0 1 -1/2 0 1/2
r1-(2/3)r2
1 0 0 7/6 2/3 -3/2
0 1 0 -1 -1 2
0 0 1 -1/2 0 1/2
左边化为单位矩阵E,所以A可逆,且 A^-1 =
7/6 2/3 -3/2
-1 -1 2
-1/2 0 1/2