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Let $k$ be an algebraically closed field of characteristic $0$. Let $X$ be a$2\times e$ matrix of linear forms over a polynomial ring $k[\mathsf{x}_1,\ldots,\mathsf{x}_n]$ (where $e,n\ge 1$). We prove that the determinantal ring$R = k[\mathsf{x}_1,\ldots,\mathsf{x}_n]/I_2(X)$ is Koszul if and only if inthe Kronecker-Weierstrass normal form of $X$, the largest length of a nilpotentblock is at most twice the smallest length of a scroll block. As anapplication, we classify rational normal scrolls whose all section rings bynatural coordinates are Koszul. This result settles a conjecture due to Conca.

Koszul determinantal rings and $2\times e$ matrices of linear forms