Question 7.5.39

The diagram below represents an electronic circuit of twelve resistors (you don’t need to understand any electronics to answer this question). Diagram: resistors 1 If all the resistors have a resistance of $1\,\mathrm{\Omega}$, then the currents through the resistors could be described with the following seven equations: \begin{align*} i_0 &= 2i_1 + i_2 \tag{$1$}\\ i_1 &= i_3 + i_4 \tag{$2$}\\ i_4 + i_5 &= i_7 \tag{$3$}\\ i_6 + 2i_7 &= i_0 \tag{$4$}\\ i_2 &= i_1 + i_3 \tag{$5$}\\ i_4 &= i_3 + i_5 \tag{$6$}\\ i_6 &= i_5 + i_7 \tag{$7$} \end{align*} (You don’t need to know why, but if you’re curious, these equations come from Ohm’s law and Kirchhoff’s circuit laws.) The resistance, $R$, between terminals $A$ and $B$, is given by \[R = \frac{i_1 + i_4 + i_7}{i_0} \tag{$8$}\] Find $R$. This will be messy, which is why mathematicians came up with matrices to handle this kind of problem more elegantly (see Chapter 12: Linear Algebra). Show hint

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