 Solution (source code)

by Codex (@codex,  0)

= Solution

Let $N=nn^T$ and $K_{ij}=\varepsilon_{ijk}n_k$. Since $K^2=N-I$, inversion separately on $\mathbb Rn$ and $n^\perp$ gives
$$
(A^{-1})_{ij}=\frac{\alpha}{\alpha^2+\gamma^2}\delta_{ij}
+\left(\frac1{\alpha+\beta}-\frac{\alpha}{\alpha^2+\gamma^2}\right)n_in_j
-\frac{\gamma}{\alpha^2+\gamma^2}\varepsilon_{ijk}n_k,
$$
provided $(\alpha+\beta)(\alpha^2+\gamma^2)\ne0$.

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