I wonder if there is a paper that can point out how to compute the determinant of a $d \times d$ autoregressive correlation matrix of the form


$$R = \begin{pmatrix}
1 & r & \cdots & r^{d-1}\\
r & 1 & \cdots & r^{d-2}\\
\vdots & \vdots & \ddots & \vdots\\ 
r^{d-1} & r^{d-2} & \cdots & 1
\end{pmatrix}$$



Determinant of correlation matrix of autoregressive model
The determinant of this matrix is $(1-r^2)^{d-1}$. This can be proven by induction on $d$ using the block matrix identity $\det \pmatrix{A & B\cr C & D} = \det(A) \det(D - C A^{-1} B)$, where $A$ is the top left $(d-1) \times (d-1)$ submatrix. Note that $$A \pmatrix{0\cr \ldots\cr 0\cr r\cr} = B$$
so $D - C A^{-1} B = 1 - r^2$.


