# Is the product of two Smith Normal Forms a the Smith normal form of the product?

Suppose A and B are square matrices of the same size over a PID R. Does the Smith Normal form of AB equal the product of the Smith normal form of A and B? I think this should be false. However, I can't quite find an example.

• It is false. Try $A = \begin{pmatrix} 1 & 0 \\ 0 & 2 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \\ 0 & 1 \end{pmatrix}$. Their Smith normal forms are both $A$, but their product's Smith normal form is not $A^2$. – darij grinberg Dec 2 '18 at 18:40
• Wow! Thanks a lot. – justanothermathstudent Dec 2 '18 at 19:28