aa= (ω^2)*(Subscript[M, 1])-2*C;
bb = C *(1+(E^(I*k*a)));
cc = C *(1+(E^(-I*k*a)));
dd= (ω^2)*(Subscript[M, 2])-2*C;

Solve[Det[(({{aa, bb}, {cc, dd}}))]==0,ω]

Plot[{(Sqrt[(1/1)+(1/0.8)-((Sqrt[1+ 0.8*(E^(-I  k)) +0.8* (E^(I k)) +0.8*0.8])/0.8)]),(Sqrt[(1/1)+(1/0.8)+((Sqrt[1+ 0.8*(E^(-I  k)) +0.8* (E^(I k)) +0.8*0.8])/0.8)])},{k,-3.14,3.14}]