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.

If finite dimension algebra A (with dim(A)= n) has an identity, then A is isomorphic with subalgebra of $\frak{B}$(A)(the algebra of bounded linear transformation on A, that is isomorphic with matrix algebras of $M_{n×n}$) under left regular representation and if A has not an identity, then left regular representarion is not necessarily injective.but if $\textit{A}_{l}$(A)=$0$ or A be an annihilator algebra then left regular representation is injective and A is isomorphic with subalgebra of matrix algebra $M_{n×n}$
and $\textit{A}_{l}$(A)={$x\in A|xA=0$}
