Let $k$ be a field and let $n\in\mathbb{N}$. Let $M_n(k)$ denote the $k$-[vector space](/page/Vector%20Space) of $n\times n$ matrices over $k$, with entrywise addition and scalar multiplication. Define matrix multiplication by the map
Then $\mu$ is $k$-bilinear, associative, and has the identity matrix $I_n$ as a two-sided multiplicative identity. Consequently $M_n(k)$ is a unital associative algebra over $k$.