# Formula for inverse of polynomial matrices like this

Suppose I have a matrix $M(n)$ defined as follows. At first, column $i$ is just the vector of $n$ copies of the variable $u_i$. Then, I multiply everything by $x$. Then I multiply the $i$-th diagonal element by $t_i$ and then I take this matrix and subtract it from the identity. For example, when $M = 4$, the result is

$\left( \begin{array}{cccc} 1-x t_1 u_1 & -x u_2 & -x u_3 & -x u_4 \\ -x u_1 & 1-x t_2 u_2 & -x u_3 & -x u_4 \\ -x u_1 & -x u_2 & 1-x t_3 u_3 & -x u_4 \\ -x u_1 & -x u_2 & -x u_3 & 1-x t_4 u_4 \\ \end{array} \right)$

Can I find a simple expression for the $i,j$ entry of the inverse of $M(n)$?