Suppose $\kappa$ is an inaccessible cardinal, and let $P$ be the $< \aleph_1-$support product of $Add(\alpha^{++}, 1)$ for singular cardinals $\alpha < \kappa.$
1- Does this forcing preserve cardinals?
2-(A weaker question) Does $\kappa$ remain inaccessible in the generic extension?
Your forcing notion will collapse all uncountable cardinals below $\kappa$ to $\aleph_1$. To see this, fix any uncountable $\gamma\lt\kappa$. Consider the first $\aleph_1$ many cardinals after $\gamma$ at which forcing occurs. For the $\xi^{th}$ such cardinal $\alpha$, we are adding a subset to $\alpha^{++}$. Consider the first ordinal in the set added at this stage. This can be any ordinal up to $\alpha^{++}$, including any ordinal below $\gamma$. Because you are using countable support, any condition specifies nontrivial sets only on a countable number of these cardinals, and so it is dense that any particular $\beta\lt\gamma$ appears at least once in such a way. Thus, in the generic extension, there will be a surjective map from $\aleph_1$ onto $\gamma$, and so $\gamma$ is collapsed.
The cardinal $\kappa$ itself is not collapsed, by a $\Delta$-system argument (and neither is any cardinal above $\kappa$ collapsed), and so this forcing makes $\kappa$ the $\aleph_2$ of the extension. In particular, it does not preserve the inaccessibility of $\kappa$.
