Let $A_1$ and $A_2$ be two $2$-by-$2$ matrices over $\mathbb{Q}$. Suppose we have the following relation $$A_1A_2=\begin{pmatrix} \frac{1}{a^2} & 0 \\ 0 & 0 \end{pmatrix},~A_2 A_1=\begin{pmatrix} 0 & 0 \\ 0 & \frac{1}{a^2} \end{pmatrix}, A_1 A_2 A_1=\begin{pmatrix} 0 & \frac{1}{a^3} \\ 0 & 0 \end{pmatrix}, ~A_2A_1A_2=\begin{pmatrix} 0 & 0 \\ \frac{1}{a^3} & 0 \end{pmatrix}, \\ A_1A_2A_1A_2=\begin{pmatrix} \frac{1}{a^4} & 0 \\ 0 & 0 \end{pmatrix}, A_2A_1A_2A_1=\begin{pmatrix} 0 & 0 \\ 0 & \frac{1}{a^4} \end{pmatrix}, A_1A_2A_1A_2A_1=\begin{pmatrix} 0 & \frac{1}{a^5} \\ 0 & 0 \end{pmatrix}, ~A_2A_1A_2A_1A_2=\begin{pmatrix} 0 & 0 \\ \frac{1}{a^5} & 0 \end{pmatrix}, ~\cdots$$ this patterns continues and here $a$ is the constant.
Is it possible to represent these values with a single general expression ?
At least can we have a general for the above $8$ expressions ?
Thanks