Consider the electrical network shown below.

The jagged lines correspond to resistors. The pair of parallel lines corresponds to a voltage source of $e$ volts. Let $I_0,I_1,I_2,I_3$ denote the input currents at each of the nodes labeled $0$ through $3,$ let $i_1,i_2,i_3,i_4,i_5,i_6$ be the currents through the various branches, as indicated, let $V_0,V_1,V_2,V_3$ be the voltages at each node and let $R_1,R_2,R_3,R_4,R_5,R_6$ be the resistance at each of the resistors in the diagram. Assume that $V_0=0.$ Finally, we set

$i=\left[\begin{array}{c}{i}_1\\ i_2\\ i_3\\ i_4\\ i_5\\ i_6\end{array}\right],I=\left[\begin{array}{c}{I}_1\\ I_2\\ I_3\end{array}\right],V=\left[\begin{array}{c}e\\ V_1\\ V_2\\ V_3\end{array}\right],e=\left[\begin{array}{cccc}{R}_1& 0& \text{c}\text{d}\text{o}\text{t}\text{s}& 0\\ 0& ?& ?& ?\\ 0& ?& ?& ?\\ 0& R_2& \text{c}\text{d}\text{o}\text{t}\text{s}& 0\\ \text{v}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}& \text{d}\text{d}\text{o}\text{t}\text{s}& \text{v}\text{d}\text{o}\text{t}\text{s}\\ 0& 0& \text{c}\text{d}\text{o}\text{t}\text{s}& 0\end{array}\right],R=$

$($a$)$ Find a matrix A such that $I=Ai.$

$($b$)$ Show that $A^{T}V=Ri-e.$

$($c$)$ Use matrix operations to eliminate i from the above two equations and thus deduce that $A{R}^{-1}{A}^{T}V=I-A{R}^{-1}e.$ Show that, when written out, this equation becomes

$\left[\begin{array}{ccc}{Y}_1+Y_3+Y_4& -Y_3& -Y_4\\ -Y_3& Y_2+Y_3+Y_5& -Y_2\\ -Y_4& -Y_2& Y_2+Y_4+Y_6\end{array}\right]\left[\begin{array}{c}{V}_1\\ V_2\\ V_3\end{array}\right]=\left[\begin{array}{c}{I}_1-Y_{1}e\\ I_2\\ I_3\end{array}\right]$

Here, $Y_i=\frac{1}{R_i}.$

$($d$)$ The system of linear equations in $($c$)$ represents the relationship between the input currents and the resulting voltages in the network. Show that this system of linear equations always has a unique solution. You need not give a formula for the solution.

$2.$ Suppose that, in problem $1,$ the resistances are: $R_1=6$ ohms, $R_2=17$ ohms, $R_3=18$ ohms, $R_4=9$ ohms, $R_5=20$ ohms, $R_6=16$ ohms and $e=15$ volts. Find the vector $V$ in terms of I.

Explain in detail part $2$ step by step on how you achieve the answer