  We approach celebrated theorems of Burnside and Wedderburn via simultaneous
triangularization. First, for a general field $F$, we prove that $M_n(F)$ is
the only irrreducible subalgebra of triangularizable matrices in $M_n(F)$
provided such a subalgebra exists. This provides a slight generalization of a
well-known theorem of Burnside. Next, for a given $n > 1$, we characterize all
fields $F$ such that Burnside's Theorem holds in $M_n(F)$, i.e., $M_n(F)$ is
the only irreducible subalgebra of itself. In fact, for a subfield $F$ of the
center of a division ring $D$, our simple proof of the aforementioned extension
of Burnside's Theorem can be adjusted to establish a Burnside type theorem for
irreducible $F$-algebras of triangularizable matrices in $M_n(D)$ with inner
eigenvalues in $F$, namely such subalgebras of $M_n(D)$ are similar to
$M_n(F)$. We use Burnside's theorem to present a simple proof of a theorem due
to Wedderburn. Then, we use our Burnside type theorem to prove an extension of
Wedderburn's Theorem as follows: A subalgebra of a semi-simple left Artinian
$F$-algebra is nilpotent iff the algebra, as a vector space over the field $F$,
is spanned by its nilpotent members and that the minimal polynomials of all of
its members split into linear factors over $F$. We conclude with an application
of Wedderburn's Theorem.
