Unless you know that some or all of your terms have finite precision, you might first try replacing all 0. with 0 and see how much smaller the result is. That should eliminate things like 0.*z^7+0.*z^8, etc.

You might then try replacing all your decimal coefficients with exact rational values. That will sometimes result in much larger expressions while it eliminates some terms that are just floating point noise. Then trying Simplify[N[...]] or Expand[N[...]] on that result might collect together some terms and get rid of large rational coefficients and possibly try to expose a linear portion of your result.

Hi Nir

Your expressions e_i which are elements inside the Matrix are linear. The determinant on the other hand consists of products, etc. of different e_i which are not linear.

Robert

Hi Nir you could try to replace all the real coefficients of, u,v,w with symbolic constants

a[1]    0 0 0 0 0 0 0 0 0 0 0
w*a[2]  0 0 0 0 0 0 0 0 0 0 0
v*a[3]  0 0 0 0 0 0 0 0 0 0 0
z*a[4]  0 0 0 0 0 0 0 0 0 0 0
w*a[5]  0 0 0 0 0 0 0 0 0 0 0
v*a[6]  0 0 0 0 0 0 0 0 0 0 0
a[7]    0 0 0 0 0 0 0 0 0 0 0
u*a[8]  0 0 0 0 0 0 0 0 0 0 0
v*a[9]  0 0 0 0 0 0 0 0 0 0 0
w*a[10] 0 0 0 0 0 0 0 0 0 0 0
a[11]   0 0 0 0 0 0 0 0 0 0 0
u*a[12] 0 0 0 0 0 0 0 0 0 0 0

Robert

0        0   0   (4*z)/3 5 (4*z)/3 0   0   5 (4*z)/3 0   0  
w*a[2]   9   0   0       0 0       0   0   0 5/2     0   0  
v*a[3]   0   9   0       0 5/2     0   0   0 0       0   0  
u*z*a[4] 4*z 4*z 9*z     0 0       0   0   0 0       0   0  
w*a[5]   0   0   0       9 0       2*z 0   0 5/2     0   0  
v*a[6]   0   0   0       4 9       5   4*z 0 0       0   0  
0        5   0   (4*z)/3 0 (4*z)/3 9*z 0   0 (4*z)/3 0   5*z
u*a[8]   0   0   5/2     0 0       2*z 9*z 0 0       0   0  
v*a[9]   0   0   0       0 5/2     0   0   9 0       2*z 0  
w*a[10]  0   0   0       0 0       0   0   4 9       5   4*z
0        0   5   (4*z)/3 0 (4*z)/3 0   5*z 0 (4*z)/3 9*z 0  
u*a[12]  0   0   5/2     0 0       0   0   0 0       2*z 9*z

If you calculate the determinant and FullSimplify you get:

27/4 z^3 (-375 + 5654 z) (v (63125 + 12 z (-15875 + 9512 z)) a[3] + 
   u (-28125 + 4 (31625 - 53286 z) z) a[4] + 
   v (169025 - 198744 z) z a[6] + 
   v (-63125 - 381390 z + 49504 z^2) a[9] + 
   w ((63125 + 12 z (-15875 + 9512 z)) a[
        2] + (-63125 - 381390 z + 49504 z^2) a[
        5] + (169025 - 198744 z) z a[10]) + 
   u (50625 + 8 z (-26175 + 13468 z)) (a[8] + a[12]))

Substitute the a with your values and it is ok, linear in u,v,w.

Robert