  In this article, the goal is to find the shortest presentation of a matrix $A
\in PSL(2,\mathbb{Z})$ in terms of the so-called continuant matrices which are
most known for their role in continued fraction theory. In chapter 7 of
arXiv:1811.01229, Morier-G\'enoud and Ovsienko investigate this problem with
the restriction that all coefficients of the continuant matrices are positive.
Now, the goal is to determine the shortest presentation allowing all integer
coefficients. To determine this minimal presentation, a few characteristic
transformations will be introduced. It will also be investigated under which
conditions such a minimal presentation becomes unique. The results are also
generalized on conjugacy classes in $PSL(2,\mathbb{Z})$.
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