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(I will write $\ell$ instead of $l$, because it confuses me less frequently.) Assuming that $\delta$ definition is a typo, and it is supposed to be $$\delta=(\ell+1)[2(\ell-1)j_{\ell-1}(kR)-kRj_\ell(kR)]$$ Your matrix's determinant is nonzero... \begin{align} \alpha\delta-\beta\gamma &= \ell \left[R j_{\ell}(kR) k - 2 j_{\ell+1}(kR) \left(\ell + ...