I encounter a situation for calculating the product of indexed matrices as follows: For $X_i,Y_i$ be any real-number indexed by $i=0,...,N-1$, consider the product $$ \prod_{i=0}^{N-1} \begin{bmatrix} 1 & X_i \\ Y_i & 0 \end{bmatrix}. $$
I was wondering if there is anything I could deduce from above one step further (e.g., bring the product into the entries or some interesting factroizations...) or is there any theory/property on the product like above I could look up. Any suggestion is appreciated.
Note that matrix multiplication being non-commutative, you need to specify the order (which $\prod$ notation doesn't do). I'll assume the $i$'s increase from left to right. Then the product is
$$ \left[ \matrix{A_N & B_N\cr C_N & D_N\cr} \right]$$ where $$ \eqalign { A_N & = A_{N-1} + B_{N-1} Y_{N-1} \cr B_N & = A_{N-1} X_{N-1}\cr C_N &= C_{N-1} + D_{N-1} Y_{N-1}\cr D_N &= C_{N-1} X_{N-1}\cr} $$ and so $$\eqalign{A_N &= A_{N-1} + A_{N-2} X_{N-2} Y_{N-1}\cr C_N &= C_{N-1} + C_{N-2} X_{N-2} Y_{N-1}\cr} $$ with $A_1 = 1$, $A_2 = 1 + X_0 Y_1$, $C_1 = C_2 = Y_0$.