<s>[INST] Inverse of matrix with very structured submatrix Does this matrix admit an easy analytic expression for its inverse?
$$\begin{bmatrix}
a_1 & 0 & 0 & 0 & 0 &0&\dots&0 \\
a_2 & 1 & -b & 0 & 0&0&\dots&0 \\
a_3 & 0 & 1 & -b & 0 & 0&\dots&0\\
a_4 & 0 & 0 & 1 & -b & 0 &\dots&0\\
\vdots \\
a_{N-1} & 0 & 0 & 0 & 0 & 0 & \dots & -b \\
a_N & 0 & 0 & 0 & 0 & 0 & \dots & 1
\end{bmatrix}$$
The special thing is that it has a very easy structure, apart from row one/column one.
 [/INST] If we call your matrix $A$, then
$$
A^{-1} = 
\begin{pmatrix}
\frac{1}{a_1} & 0 & 0 & 0 & 0 & \dots \\
-\frac{1}{a_1b^2} \sum_{k=2}^Na_kb^k & 1 & b & b^2 & b^3 & \dots \\
-\frac{1}{a_1b^3} \sum_{k=3}^Na_kb^k & 0 & 1 & b & b^2 & \dots \\
-\frac{1}{a_1b^4} \sum_{k=4}^Na_kb^k & 0 & 0 & 1 & b & \dots \\
\vdots & \vdots & & \ddots & \ddots & \\
-\frac{a_N}{a_1} & 0 & 0 & 0 & \dots & 1
\end{pmatrix},
$$
which you can verify by a direct calculation.
</s>