$1)$ If $K$ be the solution set of a system of linear equations $Ax=b$ and $K_H$ be the solution set of the corresponding homogeneous system $Ax = 0$ then for any solution $s$ to $Ax=b$:
$$K= \{s\} + K_H = \{s + k : k \in K_H\}$$
(It can be proved easily)
$2)$ Let $Ax = b$ be a system of $n$ linear equations in $n$ unknowns. If $A$ is invertible, then the system has exactly one solution, $A^{−1}b.$ and Conversely, if the system has exactly one solution, then A is invertible.
Now, answer to your question is based on the above two facts: Suppose that system has only one solution $s$ and let $K_H$ be the solution set for corresponding homogeneous system $Ax=0$. Now, according to $(1),$ $\{s\} = \{s\} + K_H$ and so, $K_H = \{0\}$ and Thus null space, $N(L_A) = \{0\}$ and hence $A$ is invertible.
