Condition on the solution of a system of linear equations to have unique variables

Question

Let $\mathbb{x}=\{x_1,x_2,\dotsc,x_n\}$ be a solution to the system of equations $A\mathbb{x=b}$. What are the conditions on $A$ and $\mathbb{b}$ so that all $x_i$'s are unique i.e. $x_i \ne x_j \;\forall\; 1\le i<j\le n$?

Results for special cases like if $A$ is invertible will also be very helpful.

Thanks.

Answer 1

Considering a non-singular $3\times3$ system, $$\begin{cases}ax+by+cy&=d\\a'x+b'y+c'y&=d'\\a''x+b''y+c''y&=d'',\end{cases}$$

the unknowns $y,z$ differ when (by Cramer)

$$\begin{vmatrix}a&d&c\\a'&d'&c'\\a''&d''&c''\end{vmatrix}\ne\begin{vmatrix}a&b&d\\a'&b'&d'\\a''&b''&d''\end{vmatrix}.$$

This can be rewritten

$$\begin{vmatrix}a&b+c&d\\a'&b'+c'&d'\\a''&b''+c''&d''\end{vmatrix}\ne0.$$

More generally, the matrix such that two columns are replaced by their sum and by the RHS vector must be non-singular.

Answer 2

For the equation to have any solution at all, $\mathbf b$ must lie in $A$’s column space. This is equivalent to the condition that $A$ and $[A\mid\mathbf b]$ have the same rank. For there to be a unique solution, $A$ must have at least as many rows as columns and have full rank.