Solving an equation involving a determinant

Question

I have a matrix which doesn't include numerical values. There are four variables in it. It is defined by

M1 = ({
      {κ0  BesselJ[m, κ0 R],
       q BesselI[m, q*R], 
       q1 BesselK[m, q1*R],
       q2 BesselK[m, q2*R]},
      {((F1 /. L -> L1) - E1 + N1*κ0^2)*BesselJ[(m + 1), κ0*R],
       ((F1 /. L -> L1) - E1 - N1*q^2)*BesselI[(m + 1), q*R],
       -((F1 /. L -> L2) - E1 - N1*q1^2)*BesselK[(m + 1), q1*R],
       -((F1 /. L -> L2) - E1 - N1*q2^2)*BesselK[(m + 1), q2*R]},
      {κ0^2 BesselJ[(m + 1),
       κ0*R], -q^2 BesselI[m + 1, q*R],
       q1^2 BesselK[(m + 1),
       q1*R], q2^2 BesselK[(m + 1), q2*R]},
      {((F1 /. L -> L1) - E1 + N1*κ0^2) κ0*BesselJ[m, κ0*R],
       ((F1 /. L -> L1) - E1 - N1*q^2) q*BesselI[m, q*R],
       ((F1 /. L -> L2) - E1 - N1*q1^2) q1*BesselK[m, q1*R],
       ((F1 /. L -> L2) - E1 - N1*q2^2) q2*BesselK[m, q2*R]}
     });

Det[M1] = 0

I want to get the formula for E1 in terms of the other variables and constants. I used Solve, but it didn't work. Can anyone please help me? Really appreciated.

Answer 1

You missed to write == and used = instead. What about

M1 = ({{\[Kappa]0 BesselJ[m, \[Kappa]0 R], q BesselI[m, q*R], 
     q1 BesselK[m, q1*R], 
     q2 BesselK[m, q2*R]}, {((F1 /. L -> L1) - E1 + N1*\[Kappa]0^2)*
      BesselJ[(m + 1), \[Kappa]0*R], ((F1 /. L -> L1) - E1 - N1*q^2)*
      BesselI[(m + 1), q*R], -((F1 /. L -> L2) - E1 - N1*q1^2)*
      BesselK[(m + 1), q1*R], -((F1 /. L -> L2) - E1 - N1*q2^2)*
      BesselK[(m + 1), 
       q2*R]}, {\[Kappa]0^2 BesselJ[(m + 1), \[Kappa]0*
        R], -q^2 BesselI[m + 1, q*R], q1^2 BesselK[(m + 1), q1*R], 
     q2^2 BesselK[(m + 1), 
       q2*R]}, {((F1 /. L -> L1) - E1 + N1*\[Kappa]0^2) \[Kappa]0*
      BesselJ[m, \[Kappa]0*R], ((F1 /. L -> L1) - E1 - N1*q^2) q*
      BesselI[m, q*R], ((F1 /. L -> L2) - E1 - N1*q1^2) q1*
      BesselK[m, q1*R], ((F1 /. L -> L2) - E1 - N1*q2^2) q2*
      BesselK[m, q2*R]}});

Solve[Det[M1] == 0, E1]