# Determinants of block tridiagonal matrices when off-diagonal blocks are not m*m matrices

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!!