I've just proved that the $QR$ factorisation is unique when the diagonal entries of $R$ are all positive. A follow up question is how many possibilities there are if this restriction is removed.
Here's what I tried:
If $A=QR$ is a $QR$ factorisation then it's clear that changing the sign of any column of $Q$ and the corresponding row of $R$ is another $QR$ factorisation. Since $Q$ and $R$ are $n\times n$ there are therefore $2^n$ possibilities with the restriction mentioned above removed, as we can either choose to switch the sign or not for each of the $n$ columns of $Q$ (and corresponding rows of $R$).
Is this a fair solution? I'm worried it doesn't take into account the chance there may be another entirely different $Q$ and $R$. How can I show there aren't other possibilities?