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First, let me give the code. These are the matrices.

gp0 = {{(1 - z^2/(4 L^2))^2/z^2, 0, 0, 0, 0, 0, 0, 
    0}, {0, (1 - z^2/(4 L^2))^2/z^2, 0, 0, 0, 0, 0, 0}, {0, 
    0, (1 - z^2/(4 L^2))^2/z^2, 0, 0, 0, 0, 0}, {0, 0, 
    0, (1 - z^2/(4 L^2))^2/z^2, 0, 0, 0, 0}, {0, 0, 0, 0, (
    1 + z^2 qb'[z]^2)/z^2, 0, 0, 0}, {0, 0, 0, 0, 0, Sin[qb[z]]^2, 0, 
    0}, {0, 0, 0, 0, 0, 0, Sin[qb[z]]^2 Sin[x1]^2, 0}, {0, 0, 0, 0, 0,
     0, 0, Sin[qb[z]]^2 Sin[x1]^2 Sin[x2]^2}};

gf0 = {{0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 
    0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, 0, 0, 0}, {0, 0, 0, 0, 0, gfzc1, 
    gfzc2, gfzc3}, {0, 0, 0, 0, gfc1z, 0, gfc1c2, gfc1c3}, {0, 0, 0, 
    0, gfc2z, gfc2c1, 0, gfc2c3}, {0, 0, 0, 0, gfc3z, gfc3c1, gfc3c2, 
    0}};

gp1 = {{0, 0, 0, 0, qb'[z] q1t, 0, 0, 0}, {0, 0, 0, 0, qb'[z] q1x, 0, 
    0, 0}, {0, 0, 0, 0, qb'[z] q1y, 0, 0, 0}, {0, 0, 0, 0, qb'[z] q1r,
     0, 0, 0}, {qb'[z] q1t, qb'[z] q1x, qb'[z] q1y, qb'[z] q1r, 
    2 qb'[z] q1z, qb'[z] q1c1, qb'[z] q1c2, qb'[z] q1c3}, {0, 0, 0, 0,
     qb'[z] q1c1, q1 Sin[2 qb[z]], 0, 0}, {0, 0, 0, 0, qb'[z] q1c2, 0,
     q1 Sin[2 qb[z]] Sin[x1]^2, 0}, {0, 0, 0, 0, qb'[z] q1c3, 0, 0, 
    q1 Sin[2 qb[z]] Sin[x1]^2 Sin[x2]^2}};

gf1 = {{0, 0, 0, 0, 0, gf1tc1, gf1tc2, gf1tc3}, {0, 0, 0, 0, 0, 
    gf1xc1, gf1xc2, gf1xc3}, {0, 0, 0, 0, 0, gf1yc1, gf1yc2, 
    gf1yc3}, {0, 0, 0, 0, 0, gf1rc1, gf1rc2, gf1rc3}, {0, 0, 0, 0, 0, 
    gf1zc1, gf1zc2, gf1zc3}, {gf1c1t, gf1c1x, gf1c1y, gf1c1r, gf1c1z, 
    0, gf1c1c2, gf1c1c3}, {gf1c2t, gf1c2x, gf1c2y, gf1c2r, gf1c2z, 
    gf1c2c1, 0, gf1c2c3}, {gf1c3t, gf1c3x, gf1c3y, gf1c3r, gf1c3z, 
    gf1c3c1, gf1c3c2, 0}};

gp2 = {{q1t^2, q1t q1x, q1t q1y, q1t q1r, q1t q1z, q1t q1c1 , 
    q1t q1c2, q1t q1c3 }, {q1x q1t, q1x^2, q1x q1y, q1x q1r, q1x q1z, 
    q1x q1c1 , q1x q1c2, q1x q1c3}, {q1y q1t, q1y q1x, q1y^2, q1y q1r,
     q1y q1z, q1y q1c1, q1y q1c2, q1y q1c3}, {q1r q1t, q1r q1x, 
    q1r q1y, q1r^2, q1r q1z, q1r q1c1, q1r q1c2, q1r q1c3}, {q1z q1t, 
    q1z q1x, q1z q1y, q1z q1r, q1z^2, q1z q1c1, q1z q1c2, 
    q1z q1c3}, {q1c1 q1t, q1c1 q1x, q1c1 q1y, q1c1 q1r, q1c1 q1z, 
    q1c1^2 + q1^2 Cos[2 qb[z]], q1c1 q1c2, q1c1 q1c3}, {q1c2 q1t, 
    q1c2 q1x, q1c2 q1y, q1c2 q1r, q1c2 q1z, q1c1 q1c2, 
    q1c2^2 + q1^2 Cos[2 qb[z]] Sin[x1]^2, q1c2 q1c3}, {q1c3 q1t, 
    q1c3 q1x, q1c3 q1y, q1c3 q1r, q1c3 q1z, q1c1 q1c3, q1c2 q1c3, 
    q1c3^2 + q1^2 Cos[2 qb[z]] Sin[x1]^2 Sin[x2]^2}};

gf2 = {{0, 0, 0, 0, 0, gf2tc1, gf2tc2, gf2tc3}, {0, 0, 0, 0, 0, 
    gf2xc1, gf2xc2, gf2xc3}, {0, 0, 0, 0, 0, gf2yc1, gf2yc2, 
    gf2yc3}, {0, 0, 0, 0, 0, gf2rc1, gf2rc2, gf2rc3}, {0, 0, 0, 0, 0, 
    gf2zc1, gf2zc2, gf2zc3}, {gf2c1t, gf2c1x, gf2c1y, gf2c1r, gf2c1z, 
    0, gf2c1c2, gf1c1c3}, {gf2c2t, gf2c2x, gf2c2y, gf2c2r, gf2c2z, 
    gf2c2c1, 0, gf2c2c3}, {gf2c3t, gf2c3x, gf2c3y, gf2c3r, gf2c3z, 
    gf2c3c1, gf2c3c2, 0}};

This is the command I want to run.

Sqrt[Det[Normal[
   Series[Normal[
     Series[(gp0 + 
         1/L^2 gf0) + δ (gp1 + 1/L^2 gf1) + δ^2 (gp2 + 
          1/L^2 gf2), {L, Infinity, 2}]], {δ, 0, 2}]]]]

The above command gives a massive, huge result.

From that result, I want is to find the terms that are independent of $L$ and the ones that go like $1/L^2$. After getting these terms, I want to find the terms that are proportional to $\delta^2$.

To that end I used, just to get the $1/L^2$ part,

Sqrt[Det[Normal[
    Series[Normal[
      Series[(gp0 + 
          1/L^2 gf0) + δ (gp1 + 1/L^2 gf1) + δ^2 (gp2 + 
           1/L^2 gf2), {L, Infinity, 2}]], {δ, 0, 2}]]]] // 
 Coefficient[#, 1/L^2] &

The result of the above is zero.

However, if I go back before I take the square root and ask for the coefficient of $\delta^4$, collect all the terms in $L$ and start deleting the powers of $L$ that I don't care about I have a non-trivial result different than zero.

This is the command I use, and I ask Mathematca to show all in order to do the manipulations.

Det[Normal[
    Series[Normal[
      Series[(gp0 + 
          1/L^2 gf0) + δ (gp1 + 1/L^2 gf1) + δ^2 (gp2 + 
           1/L^2 gf2), {L, Infinity, 2}]], {δ, 0, 2}]]] // 
  Coefficient[#, δ^4] & // Collect[#, L] &

And I don't understand why the results are different and which result I can trust. I would assume they shoud be the same.

Thanks in advance.

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It has to do with the use of the Normal command, as the following gives the correct result.

Normal[Sqrt[
     Det[Series[
       Series[(G0 + 
           1/L^2 F0) + \[Delta] (G1 + 1/L^2 F1) + \[Delta]^2 (G2 + 
            1/L^2 F2), {L, Infinity, 2}], {\[Delta], 0, 2}]]]] // 
   Collect[#, L] & // Coefficient[#, \[Delta]^2] & // Factor
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