Matrix linear algebra generators Linear algebra and special-linear group experts please help:
It is known that in principle one can generate this $C$ matrix form the $A$ and $B$ matrix below. Here
$$
C=\begin{pmatrix} 0& -1& 0\\ 1& 0& 0\\ 0& 0& 1 \end{pmatrix}
$$
from:
$$
A=\begin{pmatrix} 0& 0& 1\\ 1& 0& 0\\ 0& 1& 0 \end{pmatrix},
\text{ 
and   }\;\; 
B=\begin{pmatrix} 1& 1& 0\\ 0& 1& 0\\ 0& 0& 1 \end{pmatrix}.
$$

Question: How to generate $C$ from $A$ and $B$? i.e. So what is the exact expression to make $C=\dots A \dots B$ as a product of $A$ and $B$ matrices? As the shortest expression as possible.

p.s. I manage to generate a $C$ from a combined power of $A$ and $B$ with the exponent of 20s. I wonder what is the shortest expression?
Thank you! :o)

The expressions $A^{n_1} B^{n_2} A^{n_3} \cdots = C$ with $\sum |n_i|$ minimal are exactly the following:


*

*$B^{-1}AB^{-1}AB^{-1}A^{-1}BABAB^{-1}$

*$B^{-1}ABABA^{-1}B^{-1}AB^{-1}AB^{-1}$

*$B^{-1}A^{-1}BA^{-1}B^{-1}AB^{-1}A^{-1}BA^{-1}B^{-1}$

*$B^{-1}A^{-1}B^{-1}A^{-1}BABA^{-1}B^{-1}A^{-1}B^{-1}$


These are found by trying all such expressions ordered by $\sum |n_i|$.
