Using row transformations convert the following matrix into identity matrix

matrix is (1 -1)
(2 3)

1, -1
2, 3
-----------------------
R2 = R2 - 2 R1
1, -1
0, 5
-----------------------
R2 = R2/5
1, -1
0, 1
-----------------------
R1=R1+R2
1, 0
0, 1
-----------------------

If you do the same actions to the identity matrix, that will give you the inverse. I presume that's what you are trying to do:
1.00 -1.00 1.00 0.00
2.00 3.00 0.00 1.00 Action:
0.00 5.00 -2.00 1.00 r2 -2xr1
1.00 0.00 0.60 0.20 r1 + 1/5 r2
0.00 1.00 -0.40 0.20 r2/5

Check:
Starting matrix X Inverse matrix
1.00 0.00
0.00 1.00