  Let $\mathscr{R}$ be a finite von Neumann algebra with a faithful tracial
state $\tau $ and let $\Delta$ denote the associated Fuglede-Kadison
determinant. In this paper, we characterize all unital bijective maps $\phi$ on
the set of invertible positive elements in $\mathscr{R}$ which satisfy
$$\Delta(\phi(A)+\phi(B)) = \Delta(A+B).$$ We show that any such map originates
from a $\tau$-preserving Jordan $*$-automorphism of $\mathscr{R}$ (either
$*$-automorphism or $*$-anti-automorphism in the more restrictive case of
finite factors). In establishing the aforementioned result, we make crucial use
of the solutions to the equation $\Delta(A + B) = \Delta(A) + \Delta(B)$ in the
set of invertible positive operators in $\mathscr{R}$. To this end, we give a
new proof of the inequality $$\Delta(A+B) \ge \Delta(A) + \Delta(B),$$ using a
generalized version of the Hadamard determinant inequality and conclude that
equality holds for invertible $B$ if and only if $A$ is a nonnegative scalar
multiple of $B$.
