  Let $K$ be any field, let $L_n$ denote the Leavitt algebra of type $(1,n-1)$
having coefficients in $K$, and let ${\rm M}_d(L_n)$ denote the ring of $d
\times d$ matrices over $L_n$. In our main result, we show that ${\rm M}_d(L_n)
\cong L_n$ if and only if $d$ and $n-1$ are coprime. We use this isomorphism to
answer a question posed in \cite{PS} regarding isomorphisms between various
C*-algebras. Furthermore, our result demonstrates that data about the $K_0$
structure is sufficient to distinguish up to isomorphism the algebras in an
important class of purely infinite simple $K$-algebras.
