I want to solve equations with vector variables and vector/matrix parameters symbolically.
As a basic example I would like to be able to solve something like $[I-aG]x=b$ where $a,b \in \mathbb{R}^l$ are parameters, $x \in \mathbb{R}^l$ is the variable to be solved for, and $I$ and $G$ are $l \times l$ matrices. Namely, I would simply like Mathematica to output: $x=[I-aG]^{-1}b$.
Ultimately I would like to be able to solve symbolic systems of two linear simultaneous equations in two unknowns $x$ and $y$ such as \begin{align} Ax + By & = a \\ Cx + Dy & = b, \end{align} where all the capital objects are $l \times l$ matrices and all the lower case objects are vectors in $\mathbb{R}^l$. In this case I would like Mathematica to output something like (if my algebra is right): \begin{align} x & = (A-BD^{-1}C)^{-1}a - A^{-1}B(D-CA^{-1}B)^{-1}b \\ y & = -D^{-1}C(A-BD^{-1}C)^{-1}a + (D-CA^{-1}B)^{-1} b. \end{align}
I have been looking around here and could not really find anything addressing plain and simple algebraic manipulation of vectors and matrices. Really, the only things that differs from manipulating scalars is making sure to get the commutativity rules right, understand transposes, and write inverses rather than divisions. Is what I am talking about possible to do in Mathematica? Thank you!