(i) If $M = M^{\dagger}$ is an Hermitian $N$ dimensional matrix, $c^{\dagger}_{i}$ and $c_{i}$ $(i = 1 \ldots N)$ are creation and annihilation operators for Fermions (i.e. $ {c_{i},c^{\dagger}_{j}} = \delta_{ij}$, show that


\begin{displaymath}
c_{i} ^{\dagger} (t) = \left[ e^{-\hat M t}c^{\dagger}e^{\ha...
...^{-M} \right]_{ji}\;\;\;\;
(M = c^{\dagger}_{j} {M}_{jk}c_{k})
\end{displaymath}

(Hint consider the differential equation satisfied by $c^{\dagger}(t)$).