Lemma 11.4.10. Let $A$ be a finite central simple algebra over $k$. Then $A \otimes _ k A^{op} \cong \text{Mat}(n \times n, k)$ where $n = [A : k]$.

Proof. By Lemma 11.4.8 the algebra $A \otimes _ k A^{op}$ is simple. Hence the map

$A \otimes _ k A^{op} \longrightarrow \text{End}_ k(A),\quad a \otimes a' \longmapsto (x \mapsto axa')$

is injective. Since both sides of the arrow have the same dimension we win. $\square$

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