## The Centralizer of a Matrix is a Subspace

## Problem 660

Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define

\[W = \{ A \in V \mid AM = MA \}.\]
The set $W$ here is called the **centralizer** of $M$ in $V$.

Prove that $W$ is a subspace of $V$.

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