Suppose we have a linear system of equations given in matrix form as
$$ \begin{bmatrix} a & b & c \newline d & e & f \newline g & h & i \end{bmatrix} \begin{bmatrix} x \newline y \newline z \end{bmatrix} = \begin{bmatrix} p \newline q \newline r \end{bmatrix} \tag{1} $$ where $a, b, c, \cdots, h, i, p, q, r \in \mathbb{Z}$.
What are the conditions under which the system in Eqn. $(1)$ is Diophantine (i.e., $x,y,z$ has integer solutions)?
Here is what I came up with:
If $\Delta = \left| \begin{matrix} a & b & c \newline d & e & f \newline g & h & i \end{matrix} \right| = 1$ then $x, y, z$ are integers.
More generally, if $\Delta x = \left| \begin{matrix} p & b & c \newline q & e & f \newline r & h & i \end{matrix} \right|, \Delta y = \left| \begin{matrix} a & p & c \newline d & q & f \newline g & r & i \end{matrix} \right|, \Delta z = \left| \begin{matrix} a & b & p \newline d & e & q \newline f & h & r \end{matrix} \right|$ are all divisible by $\Delta$ then $x, y, z$ are integers.
Is there any other alternative or succint way to express this?
If $a,b,c,...i,p,q,r$ are all integers, then $x,y,z$ have integer solutions when $p$ is divisible by $ \gcd(a,b,c)$, $q$ is divisible by $ \gcd(d,e,f)$, and $r$ is divisible by $ \gcd(g,h,i)$. This condition is necessary but not sufficient.