Definition: An $n \times n$ matrix $A$ with integer entries is called unimodular if $\det A = \pm 1$.
For $n=2$, if $a, b$ are coprime integers, by Euclid's algorithm, there exist $x, y$ such that $ax-by=1$. It means matrix $$\begin{bmatrix}a&x\\b&y\end{bmatrix}$$ is an unimodular matrix. I want to do the same technique for $n=3$.
Problem: For integers $x,y,z$ such that the greatest common divisor of $x,y,z$ is 1, there exist integers $a,b,c,d,e,f$ such that $x= bf-ce, y= cd-af$ and $z= ae-bd$. Is this statement is True?
Here's one way to generate such matrices.
We can assume that $$\left\lvert\begin{array}{ccc}\pm1&x&y\\0&a&b\\0&c&d\end{array}\right\rvert = \pm 1$$ We can pick $x,y\in\mathbb Z$ arbitrarily and it suffices to guarantee that $$\left\lvert\begin{array}{cc}a&b\\c&d\end{array}\right\rvert =\pm 1.$$
Then we can do elementary row and column operations however we want and the determinant will change between $\pm 1$.