利用矩阵的行初等变换求下列矩阵的逆矩阵（帮帮忙吧）12342312111-110-2-6

Question

利用矩阵的行初等变换求下列矩阵的逆矩阵（帮帮忙吧）
1 2 3 4
2 3 1 2
1 1 1 -1
1 0 -2 -6

Answer 1

(A,E) =
1 2 3 4 1 0 0 0
2 3 1 2 0 1 0 0
1 1 1 -1 0 0 1 0
1 0 -2 -6 0 0 0 1
r1-r3,r2-2r3,r4-r3
0 1 2 5 1 0 -1 0
0 1 -1 4 0 1 -2 0
1 1 1 -1 0 0 1 0
0 -1 -3 -5 0 0 -1 1
ri-r4,i=1,2,3
0 0 -1 0 1 0 -2 1
0 0 -4 -1 0 1 -3 1
1 0 -2 -6 0 0 0 1
0 -1 -3 -5 0 0 -1 1
r1*(-1),r2+4r1,r3+2r1,r4+3r1
0 0 1 0 -1 0 2 -1
0 0 0 -1 -4 1 5 -3
1 0 0 -6 -2 0 4 -1
0 -1 0 -5 -3 0 5 -2
r2*(-1),r3+6r2,r4+5r2
0 0 1 0 -1 0 2 -1
0 0 0 1 4 -1 -5 3
1 0 0 0 22 -6 -26 17
0 -1 0 0 17 -5 -20 13
r4*(-1),交换行得
1 0 0 0 22 -6 -26 17
0 1 0 0 -17 5 20 -13
0 0 1 0 -1 0 2 -1
0 0 0 1 4 -1 -5 3
所以 A^-1 =
22 -6 -26 17
-17 5 20 -13
-1 0 2 -1
4 -1 -5 3