THERE SHOULD BE $6$ EQUATIONS FOR Loop Analysis

Consider the following circuit. The loop voltages are already labelled, please use these labels in your solution.

Answer the following questions:

$($a$).$ Identify any superloops $(1$ mark$).$ Each superloop should be identified by a comma$-$separated list of all loop currents that belong in that superloop $($for example, $"i_1,i_5,i_8",$ if loops $1,5,$ and $8$ form a superloop$).$ If the outside loop is part of a superloop it can be identified as $i_0$ or $i_{\text{o}\text{u}\text{t}\text{s}\text{i}\text{d}\text{e}}.$ If there aren't any superloops, then leave this question blank for full marks $($these circuits are randomly generated, it is possible that there aren't any in your circuit$).$

$($b$).$ Write the loop equations symbolically $(3$ marks$).$ These equations should include only the loop currents $i_n$ and any appropriate constants $($conductances$,$ resistances, voltages, currents$)$ from the circuit.

Please write everything symbolically, following the same advice given in question $1($c$).$

$($c$).$ Obtain the matrix equation $(3$ marks$).$ These matrices should be written numerically, except the unknown loop currents should be still written symbolically.

You can enter these matrices in any form you want as long as each element is separated from the other elements by some recognizable character $($comma$,$ semicolon, white space, tab, etc.$).$

$($d$).$ Solve the matrix equation to obtain the loop currents $(3$ marks$).$ These should be given in units of mA $.$ It is not necessary to write the units, unless you decide to use something other than the recommended units.

$($e$).$ Calculate the voltage $v_u$ shown on the circuit diagram $(2$ marks$).$ Your answer should be given in units of V$.$ It is not necessary to write the units, unless you decide to use something other than the recommended units.