Let $ X $ be a smooth (projective) variety and $ \mathcal{F} $ a torsion-free coherent sheaf of rank $ r $ on $ X $. The determinant $ \det \mathcal{F} $ can be defined by
(1) $ \det \mathcal F := ( \wedge^r \mathcal{F} )^{\vee \vee} $. Here the double dual denotes the reflexive hull of the rank $ 1 $ sheaf $ \wedge^r \mathcal{F} $ and therefore is a line bundle (as $ X $ is smooth).
(2) $ \det \mathcal F := \bigotimes_j (-1)^j \det E^j $ for a finite length resolution $ E^{\bullet} $ of $ \mathcal{F} $ by locally free sheaves. Here the notation $ (-1)^j \det E^j $ just means $ \det E^j $ for even $ j $ and $ (\det E^j)^{\vee} $ for odd $ j $. The determinant computed this way can be shown to be independent of the chosen complex $ E^{\bullet} $.
Are these two definitions the same? And if yes, why? I have been going crazy over this point!
