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We know that if we have the block-tridiagonal matrix $M(z)$ with blocks $A_i$, $B_i$ and $C_{i−1}$ (i =1,...,n) which are all $m \times m$ matrices $$M(z)=\begin{bmatrix} A_1 & B_1 & \cdots &\frac{1}{z}C_0\\ C_1 & \ddots & \ddots& \\ \vdots & \ddots & \ddots& B_{n-1}\\ zB_n & & C_{n-1}& A_n \end{bmatrix}$$ where the off-diagonal blocks are nonsingular, it can be associated with a transfer matrix, $$T=\begin{bmatrix} -B_n^{-1}A_n& -B_n^{-1}C_{n-1} \\ I_m & 0 \end{bmatrix} \cdots \begin{bmatrix} -B_1^{-1}A_1& -B_1^{-1}C_0 \\ I_m & 0 \end{bmatrix} $$ where $ I_m$ is the $ m×m$ unit matrix and $$\det(T)=\prod_{i=1}^\infty \det[B_i^{-1}C_{i-1}] $$

But what will the transfer matrix $T$ and the determinant $\det(T)$ be if $B_i$ and $C_{i}$ are $m \times n$ matrices ($m$ is not equal to $n$) instead of $m\times m$ matrices? What's the new formula for that?

Thank you!!

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