$($a$)$ Develop a layout for the matrix in Figure below using the $2-$opt algorithm.

$($b$)$ Identify a maximal PAG$($Planar Adjacency Matrix$).$

$($c$)$ Draw the dual of the maximal PAG identified in $($a$).$

$($d$)$ Based on the dual and using the department lengths provided in the example, develop a detailed layout by trial and error.

$($e$)$ How does it compare with the $2-$opt solution provided in $($a$)?$ Explain.

Please show all the matrices and solve them fully.