Solving nodal analysis equations using matrices- gives a full rank matrix
The schematic:



simulate this circuit &ndash; Schematic created using CircuitLab
The nodal equations:
$$\begin{array}{ccccl}
V_0\cdot \left(G_0+G_{01}\right)&amp;-V_1\cdot G_{01}&amp;+V_2\cdot 0&amp;=I\\
-V_0\cdot G_{01}&amp;+V_1\cdot \left(G_{01}+G_1+G_{12}\right) &amp;+ V_2\cdot G_{12}&amp;=0\:\text{A}\\
V_0\cdot 0 &amp;-V_1\cdot G_{12} &amp;+V_2\cdot \left(G_{12}+G_2\right) &amp;= 0\:\text{A}
\end{array}$$
Their equivalent in matrix form:
$$\left[\begin{smallmatrix}
G_0+G_{01}&amp;-G_{01}&amp;0\\
-G_{01} &amp;G_{01}+G_1+G_{12}&amp;-G_{12}\\
0 &amp;-G_{12} &amp; G_{12}+G_2
\end{smallmatrix}\right]\left[\begin{smallmatrix}V_0\\V_1\\V_2\end{smallmatrix}\right]=\left[\begin{smallmatrix}I\\0\:\text{A}\\0\:\text{A}\end{smallmatrix}\right]$$
Since \$V_0=V\$ and is known (or else \$I\$ is known and \$V\$ isn't known), you have a consistent, unique solution.
You should be able to easily spot your error.