We study the problem of polynomial identity testing (PIT) for depth

$2$ arithmetic circuits over matrix algebra. We show that identity

testing of depth $3$ ($\Sigma \Pi \Sigma$) arithmetic circuits over a

field $\F$ is polynomial time equivalent to identity testing of depth

$2$ ($\Pi \Sigma$) arithmetic circuits over

$\mathsf{U}_2(\mathbb{F})$, the algebra of upper-triangular $2\times

2$ matrices with entries from $\F$. Such a connection is a bit

surprising since we also show that, as computational models, $\Pi

\Sigma$ circuits over $\mathsf{U}_2(\mathbb{F})$ are strictly `weaker'

than $\Sigma \Pi \Sigma$ circuits over $\mathbb{F}$. The equivalence

further implies that PIT of $\Sigma \Pi \Sigma$ circuits reduces to PIT

of width-$2$ commutative \emph{Algebraic Branching

Programs}(ABP). Further, we give a deterministic polynomial time

identity testing algorithm for a $\Pi \Sigma$ circuit of size $s$ over

commutative algebras of dimension $O(\log s/\log\log s)$ over

$\F$. Over commutative algebras of dimension $\poly(s)$, we show that

identity testing of $\Pi \Sigma$ circuits is at least as hard as that

of $\Sigma \Pi \Sigma$ circuits over $\mathbb{F}$.