Artin-Wedderburn isomorphism of $\mathbb{C}[S_3]$

By the Artin - Wedderburn theorem, we know $\mathbb{C}[S_3] \cong \mathbb{C}\oplus \mathbb{C}\oplus Mat_{2x2}(\mathbb{C}) $, and the isomorphism is given by $g \mapsto (1, sgn(g), \rho(g))$, where $ \rho : \mathbb{C}[S_3] \rightarrow Mat_{2x2}(\mathbb{C})$ is a 2-dimensional irreducible representation (permutation representation modulo fixed points). Prove directly that this map is an isomorphism.


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