Be the following equation $Ax=b$ where $A$ and $b$ have entries over $\mathbb{N}$. $A$ is a full rank matrix of size say $m \times n$.
1) How to check if the equation admits a strictly positive solution in $\mathbb{(R^{*+})^{m}}$
2) How to find at least one such solution
Thanks in advance
Let $A^{\prime}$ be a matrix with $n$ rows from $A$. We choose the same $n$ rows from $b$ to get $b^{\prime}$ and solve $A^{\prime}x = b^{\prime}$. Note if $m = n$, then $A^{\prime} = A$ and $b^{\prime} = b$.
Since $A$ is of full rank, it is a one-to-one transformation. It follows that $A^{\prime}$ is also a one-to-one transformation. So the solution of $x$ will be unique. We use Cramer's rule to calculate each component of $x$.
Let $d = det(A^{\prime})$. Now $x_{i} = \frac{det(A^{\prime}_{i})}{d}$, where $A^{\prime}_{i}$ takes the $i$th column of $A^{\prime}$ and replaces it with $b^{\prime}$.