Is there any finite dimensional algebra that is not isomorphic to some algebra of matrices? Suppose that $\mathcal A$ is a finite dimensional algebra. Is it true that there always exist some isomorphism
\begin{equation}
\phi:\mathcal{A}\rightarrow C\subset M_{K\times K},
\end{equation}
where $C$ is some subalgebra of the algebra of $K\times K$ matrices? If not, what is the requirement for an algebra to be isomorphic to some algebra of matrices?
I study physics, and in general, physicists are usually interested in some algebra $\mathcal{A}$ of operators over some vector space $V$, and if $V$ is finite dimensional, the statement is obviously true. However, I do not know how to prove it for any finite dimensional algebra, and neither think of a counterexample.

In general, the answer is no. The octonions are not isomorphic to any matrix algebra because they're not associative.
If the algebra is associative then the answer is yes.
