<s>[INST] Determinant of an interesting Toeplitz matrix Let $ab=1$. Find
$$\begin{vmatrix} c & a & a^2 & ... & a^{n-1} \\ b & c & a & \dots & a^{n-2} \\ b^2 & b &  c& \dots &a^{n-3}  \\ \vdots & \vdots & \vdots & \ddots & \vdots\\ b^{n-1} & b^{n-2} & b^{n-3} & \dots & c \end{vmatrix}$$
I tried to decompose by line however it does not lead to a solution
 [/INST] Let's add a third answer for fun. Since $ab=1$, we can write
$$ \det 
\begin{pmatrix}
c&a&a^2&\cdots&a^{n-1} \\
b& c & a &\dots& a^{n-2}\\
b^2 & b & c &\cdots & a^{n-3}\\
\vdots &\vdots & \vdots & &\vdots\\
b^{n-1} & b^{n-2} &b^{n-3}& \cdots &c 
\end{pmatrix}
 =\frac{1}{b}\cdot\frac{1}{b^2}\cdots\frac{1}{b^{n-1}}\det
\begin{pmatrix}
c&1&1&\cdots&1\\
b &bc&b&\cdots &b\\
b^2 &b^2& b^2c&\cdots &b^2\\
\vdots&\vdots&\vdots& &\vdots\\
b^{n-1} & b^{n-1}& b^{n-1}&\cdots& b^{n-1}c
\end{pmatrix}\\
=\det 
\begin{pmatrix}
c & 1 & 1& \cdots & 1\\
1&c&1&\cdots&1\\
1&1&c&\cdots&1\\
\vdots &\vdots&\vdots & & \vdots \\
1&1&1&\cdots&c
\end{pmatrix}=(c-1)^{n-1}(c+n-1).
$$
where you can use any of these answers: How do I proof that the determinant of this matrix has the form $(z-1)^{n-1} (z+(n-1))$?
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