Do equivalent multiples imply equivalent matrices

Question

Let $A_1, A_2 \in \mathbb{Z}^{n \times n}$ be square matrices and assume that some multiples $C A_1, CA_2$ with $C \in \mathbb{Z}^{n \times n}$, $\mathrm{det}(C) \ne 0$ are row-equivalent i.e. $$CA_1 = U C A_2, \; \; U \in GL(n;\mathbb{Z}).$$ Does it follow that $A_1, A_2$ are also row-equivalent? or at least that they are equivalent in the weaker sense $$A_1 = P A_2 Q, \; \; P,Q \in GL(n;\mathbb{Z})?$$

If this were over a field then $C^{-1} U C$ would be an equivalence between $A_1$ and $A_2$. However $C^{-1} U C$ is not necessarily integral so that does not work here.

Answer 1

The answers to both questions are negative. Let \begin{align*} &U=\pmatrix{3&1\\ 2&1},\ C=\pmatrix{2&4\\ 2&3}, \ C^{-1}UC=\frac12\pmatrix{0&-1\\ 4&8},\\ &A_1=2C^{-1}UC=\pmatrix{0&-1\\ 4&8},\ A_2=2I. \end{align*} Then $CA_1=UCA_2$. However, $A_1$ and $A_2$ are not equivalent over $\mathbb Z$, for, if $A_1=PA_2Q=2PQ$ for some $P,Q\in GL(n;\mathbb Z)$, all entries of $A_1$ must be even. Yet this is not true.