Hi, i'm stuck on the following, please can someone help? Let $E$ be a complex holomorphic vector bundle of rank r over a compact kahler manifold $M$, let me indicate $\mathcal{E}$ the associated locally free sheaf of $E$, let $\mathcal{F}$ be a coherent subsheaf of $\mathcal{E}$ of rank $0< p < r$ such that $\frac{\mathcal{E}}{\mathcal{F}}$ is torsion free. The inclusion map $j:\mathcal{F}\rightarrow \mathcal{E}$ induces a homomorphism of sheaves $det(j): det(\mathcal{F})=({\Lambda}^p\mathcal{F})^{** } \rightarrow ({\Lambda}^p\mathcal{E})^{**}$ (with * i mean the dual sheaf), $det(j)$ is injective outside the singularity set $S_{n-1}(\mathcal{F})\subset M$ (the set of points in which $\mathcal{F}$ is not free), writing the sequence: $0 \rightarrow ker(det(j))\rightarrow det(\mathcal{F})\rightarrow (\Lambda^{p}\mathcal{E})^{ ** }$ why the sheaf $ker(det(j))$ is a torsion sheaf?

Thank you in advance