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

Thank you Robert. The determinant is linear in the elements u,v and w since they appear only in the left column. When I used rational coefficients in the entries of the matrix it works great. When I use numbers like 3.6 it gives me very long expressions.

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.

Use Chop[] This sets all values <10.^-10 to 0 This way all things still remaining like the 0.z^7+0.z^8 will go away

In[1]:= a = 10.^-21
v = a*z^7 + 0.*z^8
Chop[v]

Out[1]= 1.*10^-21

Out[2]= 0. + 1.*10^-21 z^7

Out[3]= 0