= Solution
The <antipodal map> is the restriction to the boundary of the <unit ball> of $-I$ on $\mathbb R^{n+1}$. The induced <orientation> on the <sphere> is reversed or preserved according to the <determinant> of that ambient <linear map>, namely $(-1)^{n+1}$. Its degree is therefore $(-1)^{n+1}$. With integral coefficients, for $n\geq1$ the induced top-homology map is
$$
\boxed{a_*:H_n(S^n)\cong\mathbb Z\longrightarrow\mathbb Z,\qquad m\mapsto(-1)^{n+1}m.}
$$
At $n=0$, ordinary $H_0(S^0)=\mathbb Z^2$ instead has its two generators exchanged; multiplication by $-1$ is the statement on reduced $H_0(S^0)\cong\mathbb Z$. This distinguishes the small-dimension exception from the usual top-degree formula.
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