I have a problem of the form $$\mathbf{x} = \mathbf{A}\mathbf{x} + \mathbf{b}$$ where $\mathbf{x}$ and $\mathbf{b}$ are vectors and $\mathbf{A}$ is an invertible matrix. I can solve this problem for $\mathbf{x}$ $$\mathbf{x} = (\mathbf{I}-\mathbf{A})^{-1}\mathbf{b}.$$ I'd like to try to interpret this solution by deriving a series representation for the entries of $(\mathbf{I}-\mathbf{A})^{-1}$. I think that this should be somewhat similar to solving a linear non-homogeneous ODE (e.g. page 19 here).
Some context if it is useful: The first equation comes from the market clearing conditions in an input-output model of an economy. The entries in $\mathbf{x}$ are the revenues of each firm in the economy while $\mathbf{b}$ is the vector of consumer demands (in "dollar" terms). The $\mathbf{A}$ matrix contains the elasticity/share parameters for the inputs of the firms. Finally, $(\mathbf{I}-\mathbf{A})^{-1}$ is known as the Leontief Inverse. There is a textbook explanation of this matrix that has to do with the amount of good i needed to produce an additional unit of good j (loosely put). My idea is to use the series representation to show this explicitly.