I need some help on this since I clearly am not confident with Gaussian functional integration. Can anyone explain where the marked identities on the picture hold? I assume you have to use some formulas for Gaussian functional integration, but the only thing I could find is the one given in the bottom of the picture and that is not generalized to matrices and anyway it does not match with the results marked. Where does the determinant of A go, for example? Is it absorbed into the measure and in general which things can be absorbed into the measure, and what does it even to absorb something into a measure in functional integration?