The matrix is
\begin{equation*}
M =
\begin{pmatrix}
a_1b_1+1 & a_1b_2 & \cdots & a_1b_n \\
a_2b_1 & a_2b_2+1 & \cdots & a_2b_n \\
\vdots & \vdots & \ddots & \vdots \\
a_nb_1 & a_{m,2} & \cdots & a_nb_n+1
\end{pmatrix}
\end{equation*}
This matrix is decomposable to be the sum of identity matrix plus $\alpha^T \beta$, where $\alpha=(a_1,\ldots,a_n)$, $\beta=(b_1,\ldots,b_n)$. But I am not sure what can I do from here. Any hint/comment is welcome!
Best Answer
\begin{align*} (I + uv^\top)^{-1} &= I - uv^\top + (uv^\top)^2 - (uv^\top)^3 + \cdots \\ &= I - u(1 - v^\top u + (v^\top u)^2 - \cdots)v^\top \\ &= I - u \left(\frac{1}{1 + v^\top u} \right) v^\top \\ &= I - \frac{1}{1 + v^\top u} u v^\top. \end{align*}
This is a special case of the Sherman-Morrison Formula.