= Solution
Choose the ordered <angular momentum eigenstates> $(|1,1\rangle,|1,0\rangle,|1,-1\rangle)$, so $J_3=\hbar\operatorname{diag}(1,0,-1)$. Since $[J_3,J_\pm]=\pm\hbar J_\pm$, a nonzero <matrix element> of $J_+$ must increase the <magnetic quantum number> by one, and one of $J_-$ must decrease it by one. Thus only entries $(1,2),(2,3)$ of $J_+$ and $(2,1),(3,2)$ of $J_-$ can be nonzero. The <adjoint operator> identity $J_-=J_+^\dagger$ gives
$$
\boxed{J_+=\begin{pmatrix}0&A&0\\0&0&B\\0&0&0\end{pmatrix},\qquad
J_-=\begin{pmatrix}0&0&0\\\overline A&0&0\\0&\overline B&0\end{pmatrix}.}
$$
For the $j=1$ <irreducible spin representation>, both adjacent transitions are present; the following calculation determines their sizes.
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