Let $M_n(\mathbb{C})$ be the algebra of $n\times n$ matices over $\mathbb{C}$. By Skolem-Noether, all automorphisms of this algebra are inner. What is the group of automorphisms of $M_n(\mathbb{C})$?
2026-03-25 12:29:33.1774441773
automorphism group of a matrix algebra
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What you said says that the homomorphism from $GL_n(\mathbb C)\to Aut(M_n(\mathbb C))$ is surjective.
Think about what the kernel is and apply the first isomorphism theorem.