Y-Δ transform with generators. Equivalence of three-terminal devices
I believe the reason you are having a zero determinant is because the transformation, if it exists, is not unique. 

Firstly, I'd like to redefine \$I_b\$ in the opposite orientation to restore the symmetry (It's not really a big deal though.)

Thevenin's Theorem says that when we look at the circuit from any 2 terminals, say \$A\$ and \$B\$, it will be indistinguishable from a resistor connected in series to a voltage source. 

We can measure the Thevenin's resistance by using a capacitor and observing it's time constant. The transformation from \$R\$'s to \$G\$'s should therefore be identical to the Y-\$\Delta\$ transform. 

The problem arises when we try to solve for the currents \$I_a\$, \$I_b\$ and \$I_c\$. Suppose that \$(I_a, I_b, I_c)\ = (I_1, I_2, I_3)\$ is a solution that gives the correct open-circuit voltage between all possible pairs of terminals (i.e. \$AB, AC, BC\$). Then you can verify that \$(I_a, I_b, I_c)\ = (I_1+\Delta I, I_2+\Delta I, I_3+\Delta I)\$ will also be a solution for any \$\Delta I\$.