Inverse mapping (affine transformation) for singular transform matrix
Usually, when an inverse is required but the matrix you are taking the inverse of is not guaranteed to be invertible, then the (Moore-Penrose) pseudoinverse is used:

In R, if $T$ is

T
        [,1] [,2] [,3]
[1,]    1    1    0
[2,]    1    1    0
[3,]    0    0    1


Then the pseudoinverse calculated using ginv is

ginv(T)
     [,1] [,2] [,3]
[1,] 0.25 0.25    0
[2,] 0.25 0.25    0
[3,] 0.00 0.00    1


Clearly, this won't give the identity matrix when multiplying $T$, but it's as close as you can get with $T$ singular.

T*ginv(T)
     [,1] [,2] [,3]
[1,] 0.25 0.25    0
[2,] 0.25 0.25    0
[3,] 0.00 0.00    1