Question

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.

Answer 1

(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