Fast Computation of Sparse Symbolic Determinants 

In[1]:=

X m = {{1/Subscript[e, 1], 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/ Subscript[e, 2], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 2], 0, 0, 0, Subscript[ e, 2], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/ Subscript[e, 2], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 2], 0, 0, 0, Subscript[ e, 2], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 2], 0, 0, 0, Subscript[ e, 2], 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/ Subscript[e, 2], 0, 0, 0, Subscript[e, 2], 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 2], 0, 0, 0, Subscript[ e, 2], 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/ Subscript[e, 2], 0, 0, 0, Subscript[e, 2]}, {Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 1/ Subscript[e, 2], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 1/Subscript[e, 2], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 1/ Subscript[e, 2], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 1/Subscript[e, 2], 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 1/Subscript[e, 2], 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 1/ Subscript[e, 2], 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 1/Subscript[e, 2], 0}, {0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Subscript[e, 1], 0, 0, 0, 1/Subscript[e, 1], 0, 0, 0, Subscript[e, 2], 0, 0, 0, 1/ Subscript[e, 2]}, {((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 1] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\))/Subscript[e, 2], 0, 0, 0, Subscript[e, 2] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)), 0, 0, 0, (\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 1])/Subscript[ e, 1], 0, 0, 0, -\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 1], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 1])/Subscript[ e, 2], 0, 0, 0, -\[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 1], 0, 0, 0}, {0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 1] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\))/Subscript[e, 2], 0, 0, 0, Subscript[e, 2] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)), 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 1])/ Subscript[e, 1]), 0, 0, 0, \[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 1], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 1])/ Subscript[e, 2]), 0, 0, 0, \[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 1], 0, 0}, {0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 1] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\))/Subscript[e, 2], 0, 0, 0, Subscript[e, 2] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)), 0, 0, 0, (\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 2])/Subscript[ e, 1], 0, 0, 0, -\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 2], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 2])/Subscript[ e, 2], 0, 0, 0, -\[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 2], 0}, {0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 1] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\))/Subscript[e, 2], 0, 0, 0, Subscript[e, 2] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)), 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 2])/ Subscript[e, 1]), 0, 0, 0, \[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 2], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 2])/ Subscript[e, 2]), 0, 0, 0, \[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 2]}, {(\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 1])/ Subscript[e, 1], 0, 0, 0, -\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 1], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 1])/Subscript[ e, 2], 0, 0, 0, -\[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 1], 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 1] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\))/Subscript[e, 2], 0, 0, 0, Subscript[e, 2] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\)), 0, 0, 0}, {0, -((\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 1])/ Subscript[e, 1]), 0, 0, 0, \[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 1], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 1])/ Subscript[e, 2]), 0, 0, 0, \[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 1], 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 1] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\))/Subscript[e, 2], 0, 0, 0, Subscript[e, 2] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\)), 0, 0}, {0, 0, (\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 2])/Subscript[ e, 1], 0, 0, 0, -\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 2], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 2])/Subscript[ e, 2], 0, 0, 0, -\[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 2], 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 1] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))/Subscript[e, 2], 0, 0, 0, Subscript[e, 2] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\)), 0}, {0, 0, 0, -((\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 2])/ Subscript[e, 1]), 0, 0, 0, \[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 2], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 2])/ Subscript[e, 2]), 0, 0, 0, \[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 2], 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 1] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))/Subscript[e, 2], 0, 0, 0, Subscript[e, 2] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))}, {Subscript[e, 1] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 2] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\))/Subscript[e, 2], 0, 0, 0, \[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 1], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 1])/ Subscript[e, 1]), 0, 0, 0, \[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 1], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 1])/ Subscript[e, 2]), 0, 0, 0}, {0, Subscript[e, 1] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 2] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\))/Subscript[e, 2], 0, 0, 0, -\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 1], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 1])/Subscript[ e, 1], 0, 0, 0, -\[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 1], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 1])/Subscript[ e, 2], 0, 0}, {0, 0, Subscript[e, 1] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 2] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\))/Subscript[e, 2], 0, 0, 0, \[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 2], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 2])/ Subscript[e, 1]), 0, 0, 0, \[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 2], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 2])/ Subscript[e, 2]), 0}, {0, 0, 0, Subscript[e, 1] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 2] ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)), 0, 0, 0, ((1 + \[Mu]) \!\(\*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)]\) + \!\(\*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\))/Subscript[e, 2], 0, 0, 0, -\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 2], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 2])/Subscript[ e, 1], 0, 0, 0, -\[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 2], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 2])/Subscript[ e, 2]}, {\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 1], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 1])/ Subscript[e, 1]), 0, 0, 0, \[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 1], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 1])/ Subscript[e, 2]), 0, 0, 0, Subscript[e, 1] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 2] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\))/Subscript[e, 2], 0, 0, 0}, {0, -\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 1], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 1])/Subscript[ e, 1], 0, 0, 0, -\[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 1], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 1])/Subscript[ e, 2], 0, 0, 0, Subscript[e, 1] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 2] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(1\), \(2\)]\)\))/Subscript[e, 2], 0, 0}, {0, 0, \[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 2], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 2])/ Subscript[e, 1]), 0, 0, 0, \[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 2], 0, 0, 0, -((\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 2])/ Subscript[e, 2]), 0, 0, 0, Subscript[e, 1] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 2] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))/Subscript[e, 2], 0}, {0, 0, 0, -\[Mu] Subscript[e, 1] Subscript[\[Alpha], 1] Subscript[\[Beta], 2], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 1] Subscript[\[Beta], 2])/Subscript[ e, 1], 0, 0, 0, -\[Mu] Subscript[e, 2] Subscript[\[Alpha], 2] Subscript[\[Beta], 2], 0, 0, 0, (\[Mu] Subscript[\[Alpha], 2] Subscript[\[Beta], 2])/Subscript[ e, 2], 0, 0, 0, Subscript[e, 1] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(1\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))/Subscript[e, 1], 0, 0, 0, Subscript[e, 2] (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\)), 0, 0, 0, (\!\( \*SubsuperscriptBox[\(\[Alpha]\), \(2\), \(2\)] + \(\((1 + \[Mu])\)\ \*SubsuperscriptBox[\(\[Beta]\), \(2\), \(2\)]\)\))/Subscript[e, 2]}};

This defines a 32×32 symbolic matrix reported by a Mathematica user. The determinant computation would not finish in two days on a large-memory machine. It now finishes in less than 100 milliseconds on a normal machine.

The matrix is sparse, but contains general formulas.

In[2]:=

X MatrixPlot[m]

Out[2]=

In[3]:=

X MatrixForm[m[[28 ;; 32, 28 ;; 32]]]

Out[3]//MatrixForm=

Computing the determinant is now fast.

In[4]:=

X det = Det[m]; // Timing

Out[4]=

In[5]:=

X LeafCount[det]

Out[5]=