A connected resistive network has three essential nodes labelled A, B, and N

A connected resistive network has three essential nodes labelled \(A\), \(B\), and \(N\). Take \(N\) as the reference (ground) node, and let \(v_A\) and \(v_B\) be the voltages of \(A\) and \(B\) with respect to \(N\). The branches are as follows: - a \(2\,\Omega\) resistor between \(A\) and \(N\); - a \(4\,\Omega\) resistor between \(A\) and \(B\); - a \(4\,\Omega\) resistor between \(B\) and \(N\); - an independent current source of \(3\,\mathrm{A}\) connected between \(N\) and \(A\), with its reference arrow directed from \(N\) toward \(A\); - an independent current source of \(1\,\mathrm{A}\) connected between \(N\) and \(B\), with its reference arrow directed from \(N\) toward \(B\). Using systematic nodal analysis, write the two independent KCL equations in conductance matrix form \(G\begin{bmatrix}v_A\\v_B\end{bmatrix}=i\). State every entry of \(G\) in siemens and every entry of \(i\) in amperes, with the unknown vector ordered as \((v_A,v_B)\). Solve for \(v_A\) and \(v_B\) in volts. Then compute the current through the \(4\,\Omega\) resistor between \(A\) and \(B\), taken positive from \(A\) toward \(B\).

Change every resistance into a conductance and treat each current-source arrow as an injection into the node it points toward.