# Solving an equation involving a determinant

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.

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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]

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I tried with that too. And end up with this error. "Solve::inex: Solve was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Solve require exact input, providing Solve with an exact version of the system may help. >>" –  TMH Mar 6 '13 at 0:17
I can give m=0,R=100. Even then it doesn't work. –  TMH Mar 6 '13 at 0:19
What I actually want is to find E1 values when Determinant of matrix =0. Anyone, please help me. –  TMH Mar 6 '13 at 0:20
@Thakshila It seems you have assigned values to some of the variables. Can you restart the Kernel or Mathematica and try again? –  halirutan Mar 6 '13 at 2:29
Thanks for helping me halirutan. But, it seems to be not working. Because it is still running. I got another idea to get a solution for my problem. I know E1 range. So, I will make tables for E1 and Det[M1]. Then how can get E1, when P1=0? Do you have any idea about that? –  TMH Mar 6 '13 at 3:52