Proof involving exponential of anticommuting operators

Joker93
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Homework Statement
The problem is to prove equation (5.31) from the book "Quarks, gluons and lattices" by Creutz. It involves anticommuting operators and functions of them acting on defined states, which I give below.
Relevant Equations
On page 23 of the book "Quarks, gluons and lattices" by Creutz, he defines a state
$$\langle\psi|=\langle 0|e^{bFc}e^{\lambda b^\dagger G c^\dagger}$$
where ##\lambda## is a number, ##F, G## are ##N\times N## symmetric matrices and ##b, c## are vectors whose components ##b_m, c_m## are operators such that their anticommutators satisfy
$$\{b_m^\dagger, b_n\}=\{c_m^\dagger, c_n\}=\delta_{mn}$$
with every other anti-commutator being zero, and the state ##\langle 0 |## such that
$$\langle 0 |c^\dagger_m=\langle 0 |b^\dagger_m=0$$
Creutz says that a straightfoward calculation can lead us to proving that
$$\langle\psi|b^\dagger=-\langle\psi|(F^{-1}-\lambda G)^{-1} c=-\langle\psi|(1-\lambda FG)^{-1}Fc$$
For ##N=1##, I have managed to prove this, but for ##N>1##, I am struggling with how to show this. Something that I managed to prove is that
$$\langle\psi |b_k^\dagger=-\langle 0 | \sum_{n=1}^N F_{kn}c_n \prod_{m=1\neq k, l}^N \left(1+b_m F_{ml}c_l \right)$$
which generalizes what I initially found for the ##N=1## case, which is ##\langle\psi|b^\dagger=-\langle 0|Fc##. For this last result for the ##N=1## case, I then substituted ##\langle 0 |=\langle \psi | e^{-\lambda b^\dagger G c^\dagger}e^{-bFc}##, and after some manipulations, I reached the end of the proof. Doing something similar for the ##N>1## case, I found it too difficult to get to a meaningful result (or even close to the final result).

Note that for the ##N>1## case, I have used that, for example,
$$e^{bFc}=e^{\sum_{ij}b_i F_{ij} c_j}=\prod_{ij}(1+b_iF_{ij}c_j)$$
which is found on page 193 of Fradkin's book "Quantum Field theory: an integrated approach". Note that there, Fradkin was talking about Grassmann variables being on the exponential, whereas here we have the above non-trivial anticommutation relations; so, this might have been where I got it wrong.

If anybody can give a hint or some guidance on this, or even provide with some reference that can help, it would be greatly appreciated.
 
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Note that I have proved that the last relation holds.
If anybody could help with the problem by just even a suggestion, It would be appreciated.
 
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