Article with Investigation

Hi,

I have these two sentences:

1. Old Mrs Smith was found dead. An investigation into her death concluded that she'd had a heart attack.

2. Old Mrs Smith was found dead. According to the investigation into her death, she'd had a heart attack.

I used "an" in #1 because I am saying: There was an investigation, and it concluded . . .

In #2, I assume that the context makes it obvious what investigation I am referring (the one into her death and it was the only one).

I could just as easily switch the articles in #1 and #2 and it would be correct, I think. Do I have it correct?