PLP= { \
{  0,  b,    a,  0,    0,  0,    0,  0 }, \
{ -b,  0,    0, -a,    0,  0,    0,  0  }, \
\
{ -a,  0,    0,  b,    a,  0,    0,  0  }, \
{  0,  a,   -b,  0,    0, -a,    0,  0  }, \
\
{  0,  0,   -a,  0,    0,  b,    a,  0  }, \
{  0,  0,    0,  a,   -b,  0,    0, -a  }, \
\
{  0,  0,    0,  0,   -a,  0,    0,  b  }, \
{  0,  0,    0,  0,    0,  a,   -b,  0  } \
};
Print["det(PLP)"]
Simplify[Det[PLP]]
Print["(PLP)^-1det(PLP)"]
Inverse[PLP]*Det[PLP]

Print["det(PLP)"]
TeXForm[Simplify[Det[PLP]]]
Print["det(-PLP)"]
TeXForm[Simplify[Det[-PLP]]]
Print["(PLP)^-1det(PLP)"]
TeXForm[Inverse[PLP]*Det[PLP]]