A={{a,0},{0,-a}};
B={{0,b},{-b,0}};
M = -Transpose[A] + c*B + c^2*A;
Print["M = "]; TeXForm[Expand[M]]
Print["\\det M = "]; TeXForm[Expand[Det[M]]]

d = Det[M];
aa = Coefficient[d*c, c^5];
bb = Coefficient[d*c, c^3];
cc = Coefficient[d*c, c];

c1 =  Sqrt[(-bb + Sqrt[bb^2 - 4 aa cc]) / (2 aa)];
c2 =  Sqrt[(-bb - Sqrt[bb^2 - 4 aa cc]) / (2 aa)];
c3 = -Sqrt[(-bb + Sqrt[bb^2 - 4 aa cc]) / (2 aa)];
c4 = -Sqrt[(-bb - Sqrt[bb^2 - 4 aa cc]) / (2 aa)];

Print["c_1 = "]; TeXForm[Simplify[c1]]
Print["c_2 = "]; TeXForm[Simplify[c2]]
Print["c_3 = "]; TeXForm[Simplify[c3]]
Print["c_4 = "]; TeXForm[Simplify[c4]]

M1 = Simplify[-Transpose[A] + c1*B + c1^2*A];
M2 = Simplify[-Transpose[A] + c2*B + c2^2*A];
M3 = Simplify[-Transpose[A] + c3*B + c3^2*A];
M4 = Simplify[-Transpose[A] + c4*B + c4^2*A];

Print["M_1 = "]; TeXForm[M1]
Print["M_2 = "]; TeXForm[M2]
Print["M_3 = "]; TeXForm[M3]
Print["M_4 = "]; TeXForm[M4]

Print["det M_1 = "]; TeXForm[Simplify[Det[M1]]]
Print["det M_2 = "]; TeXForm[Simplify[Det[M2]]]
Print["det M_3 = "]; TeXForm[Simplify[Det[M3]]]
Print["det M_4 = "]; TeXForm[Simplify[Det[M4]]]

Print["Row reds:"]

TeXForm[Simplify[RowReduce[M1]]]
TeXForm[Simplify[RowReduce[M2]]]
TeXForm[Simplify[RowReduce[M3]]]
TeXForm[Simplify[RowReduce[M4]]]

Print["Ranks:"]
TeXForm[Simplify[MatrixRank[M1]]]
TeXForm[Simplify[MatrixRank[M2]]]
TeXForm[Simplify[MatrixRank[M3]]]
TeXForm[Simplify[MatrixRank[M4]]]

(*
Mma just gives the zero solution, even if more are possible ...
z={{0},{0}};
Print["Solves:"]
TeXForm[LinearSolve[M1,z]]
TeXForm[LinearSolve[M2,z]]
TeXForm[LinearSolve[M3,z]]
TeXForm[LinearSolve[M4,z]]
*)