ϕ=Pi/4;
J={{Cos[ϕ],-ρ*Sin[ϕ],0},{Sin[ϕ],ρ*Cos[ϕ],0},{0,0,1}};
Jt=Transpose[J];
invJ=Inverse[J];
invJt=Transpose[invJ];
detJ=ρ;
Λ={{a,0,0},{0,b,0},{0,0,c}};
ϵ={{ϵr,0,0},{0,ϵr,0},{0,0,ϵr}};
ϵc1=Simplify[Dot[invJt,invJ]*detJ*ϵr]
ϵd1=Simplify[Dot[Dot[J,ϵc1],Jt]/detJ]
ϵc=Simplify[Expand[Dot[Dot[invJt,ϵ],invJ]*detJ]];
ϵd=Simplify[Expand[Dot[Dot[J,ϵc],Jt]/detJ]];
out2=Λ*out1;
out3=Simplify[J*out2*Jt/detJ];