The determinant is of the first degree with respect to f. If you insist on solving for x it may be unfeasible. Here is a numeric case:

det = Det[mat] // Simplify // Factor
det1 = Simplify[det /. {a -> -1/2, b -> 1/4, c -> 1, f -> 1}]
Plot[det1, {x, 0, 1}]
Solve[det1 == 0 && 0 < x < 1]

You can try. Good luck!

Yes, I would like to have the solution with all parameters.The solution can be complicated. Anyway, thank you very much for your help!!

Thank you! Your method gives the correct result. However, when I change some parameters to get a new matrix, the solution for x becomes a complex transcendental equation, and it is not possible to get a solution for x. How do I find the solution X to the system of linear equations in this case? The parameters also satisfy -1 < a < 1, -0.5 < b < 0.5, c > 0, f ≠ 0, and 0 < x < 1. Here is the new matrix:

mat={{(-2 + x) Cos[1/2 \[Pi] (-2 + x)] - 
   f (-2 + x) Sin[1/2 \[Pi] (-2 + x)], -f (-2 + x) Cos[
     1/2 \[Pi] (-2 + x)] + (2 - x) Sin[1/2 \[Pi] (-2 + x)], 
  x Cos[(\[Pi] x)/2] - 
   f (-2 + x) Sin[(\[Pi] x)/2], -f (-2 + x) Cos[(\[Pi] x)/2] - 
   x Sin[(\[Pi] x)/2], 0, 0, 0, 
  0}, {(-2 + x) Cos[1/2 \[Pi] (-2 + x)], (2 - x) Sin[
    1/2 \[Pi] (-2 + x)], 
  x Cos[(\[Pi] x)/2], -x Sin[(\[Pi] x)/2], (2 - x) Cos[
    3/2 \[Pi] (-2 + x)], (2 - x) Sin[3/2 \[Pi] (-2 + x)], -x Cos[(
    3 \[Pi] x)/2], -x Sin[(3 \[Pi] x)/2]}, {1/
   2 (-2 + a c - b c) (-2 + x) Cos[1/2 \[Pi] (-2 + x)], 
  1/2 (2 - a c + b c) (-2 + x) Sin[1/2 \[Pi] (-2 + x)], 
  1/2 (-2 + a c - b c) (-1 - (-2 + c - b c)/(2 - a c + b c) + 
     x) Cos[(\[Pi] x)/2], 
  1/2 (2 - a c + b c) (-1 - (-2 + c - b c)/(2 - a c + b c) + 
     x) Sin[(\[Pi] x)/2], (-2 + x) Cos[
    3/2 \[Pi] (-2 + x)], (-2 + x) Sin[
    3/2 \[Pi] (-2 + x)], (-1 + 1/2 (2 - c - a c) + x) Cos[(3 \[Pi] x)/
    2], (-1 + 1/2 (2 - c - a c) + x) Sin[(3 \[Pi] x)/2]}, {Sin[
   1/2 \[Pi] (-2 + x)], Cos[1/2 \[Pi] (-2 + x)], Sin[(\[Pi] x)/2], 
  Cos[(\[Pi] x)/2], Sin[3/2 \[Pi] (-2 + x)], -Cos[3/2 \[Pi] (-2 + x)],
   Sin[(3 \[Pi] x)/2], -Cos[(3 \[Pi] x)/2]}, {0, 1, 0, 1, 0, -1, 
  0, -1}, {-2 + x, 0, x, 0, 2 - x, 0, -x, 0}, {0, 
  1/2 (2 - a c + b c) (-2 + x), 0, 
  1/2 (2 - a c + b c) (-1 + (-2 + c - b c)/(2 - a c + b c) + x), 0, 
  2 - x, 0, 1 + 1/2 (2 - c - a c) - x}, {1/2 (2 - a c + b c) (-2 + x),
   0, 1/2 (2 - a c + b c) (-1 - (-2 + c - b c)/(2 - a c + b c) + x), 
  0, 2 - x, 0, 1 + 1/2 (-2 + c + a c) - x, 0}}

Attachments:

1-1.nb