Represent a row or column vector in matrix algebra

Question

I just started to learn matrix algebra. Using the properties has been straight forward but recently I've encountered a hiccup. That is, how do I represent a column vector with i rows and 1 column in the following equation?

$e^TA^{-1}(y+x) = c$

where A is a i by i matrix, x and y are column vector with i rows, c is a constant and e is column vector of 1's with i rows. I want to represent x in other terms.

I can get it to look like:

$e^TA^{-1}x = c - e^TA^{-1}y$

but I don't know how to deal with the $e^TA^{-1}$ term since we can't just divide it.

Answer 1

$x$ can't be uniquely determine.

Let $\color{blue}{x=Az}$, then we have

$$e^TA^{-1}(Az)=c-e^TA^{-1}y$$

$$\color{blue}{e^Tz=c-e^TA^{-1}y}$$

We just require the sum of components of $z$ to be equal to $c-e^TA^{-1}y$.

For example, a particular solution can be $x=A\left(\frac{c-e^TA^{-1}y}{i}e\right)=\left(\frac{c-e^TA^{-1}y}{i}\right)(Ae)$

Answer 2

For convenience, let's define the variables $$\eqalign{ &a^T = e^TA^{-1} \cr &\lambda = c-a^Ty \cr }$$ You want to solve $$a^Tx=\lambda$$ but there isn't a unique solution.

For any vector $b$ that's not orthogonal to $a$, setting $$x=\frac{\lambda b}{a^Tb}$$ will satisfy the equation.