Inverse with determinant and adjoint functions:


(11:24) gp > A = [1,2,0;3,-1,1;0,-2,2]
%1 = 
[1 2 0]

[3 -1 1]

[0 -2 2]

(11:24) gp > B = (1/matdet(A))*matadjoint(A)
%3 = 
[0 1/3 -1/6]

[1/2 -1/6 1/12]

[1/2 -1/6 7/12]

(11:26) gp > A*B
%4 = 
[1 0 0]

[0 1 0]

[0 0 1]

(11:26) gp > B*A
%5 = 
[1 0 0]

[0 1 0]

[0 0 1]


Inverse with row reduction:


(11:28) gp > E=[1,0,0;0,1,0;0,0,1]
%9 = 
[1 0 0]

[0 1 0]

[0 0 1]

(11:28) gp > AE=concat(A,E)
%10 = 
[1 2 0 1 0 0]

[3 -1 1 0 1 0]

[0 -2 2 0 0 1]

(11:28) gp > E1=[1,0,0;-3,1,0;0,0,1]
%11 = 
[1 0 0]

[-3 1 0]

[0 0 1]

(11:29) gp > A1=E1*AE
%12 = 
[1 2 0 1 0 0]

[0 -7 1 -3 1 0]

[0 -2 2 0 0 1]

(11:29) gp > E2 = [1,0,0;0,-1/7,0;0,0,1]
%13 = 
[1 0 0]

[0 -1/7 0]

[0 0 1]

(11:30) gp > A2=E2*A1
%14 = 
[1 2 0 1 0 0]

[0 1 -1/7 3/7 -1/7 0]

[0 -2 2 0 0 1]

(11:30) gp > E3=[1,-2,0;0,1,0;0,0,1]
%15 = 
[1 -2 0]

[0 1 0]

[0 0 1]

(11:31) gp > A3=E3*A2
%16 = 
[1 0 2/7 1/7 2/7 0]

[0 1 -1/7 3/7 -1/7 0]

[0 -2 2 0 0 1]

(11:31) gp > E4=[1,0,0;0,1,0;0,2,1]
%18 = 
[1 0 0]

[0 1 0]

[0 2 1]

(11:32) gp > A4=E4*A3
%19 = 
[1 0 2/7 1/7 2/7 0]

[0 1 -1/7 3/7 -1/7 0]

[0 0 12/7 6/7 -2/7 1]


(11:35) gp > E5=[1,0,0;0,1,0;0,0,7/12]
%25 = 
[1 0 0]

[0 1 0]

[0 0 7/12]

(11:35) gp > A5=E5*A4
%26 = 
[1 0 2/7 1/7 2/7 0]

[0 1 -1/7 3/7 -1/7 0]

[0 0 1 1/2 -1/6 7/12]


(11:37) gp > E6=[1,0,0;0,1,1/7;0,0,1]
%31 = 
[1 0 0]

[0 1 1/7]

[0 0 1]

(11:38) gp > A6=E6*A5
%32 = 
[1 0 2/7 1/7 2/7 0]

[0 1 0 1/2 -1/6 1/12]

[0 0 1 1/2 -1/6 7/12]


(11:39) gp > E7=[1,0,-2/7;0,1,0;0,0,1]
%39 = 
[1 0 -2/7]

[0 1 0]

[0 0 1]

(11:39) gp > A7=E7*A6
%40 = 
[1 0 0 0 1/3 -1/6]

[0 1 0 1/2 -1/6 1/12]

[0 0 1 1/2 -1/6 7/12]



compare to:


(11:24) gp > B = (1/matdet(A))*matadjoint(A)
%3 = 
[0 1/3 -1/6]

[1/2 -1/6 1/12]

[1/2 -1/6 7/12]