Assume that $n$ is a multiple of 3, the other two cases should be similar.For example, $n=6$
$$\tag{1}\label{1}
A = \begin{pmatrix}
a_1 & 1 & 0 & 0 & 0 & 0 \\
a_2 & 0 & 1 & 0 & 0 & 0 \\
a_3 & 0 & 0 & 1 & 0 & 0 \\
a_4 & 0 & 0 & 0 & 1 & 0 \\
a_5 & 0 & 0 & 0 & 0 & 1 \\
a_n & 0 & 0 & 0 & 0 & 0
\end{pmatrix},\quad
A^{-1} = \begin{pmatrix}
0 & 0 & 0 & 0 & 0 & 1/a_n \\
1 & 0 & 0 & 0 & 0 & -a_1/a_n \\
0 & 1 & 0 & 0 & 0 & -a_2/a_n \\
0 & 0 & 1 & 0 & 0 & -a_3/a_n \\
0 & 0 & 0 & 1 & 0 & -a_4/a_n \\
0 & 0 & 0 & 0 & 1 & -a_5/a_n