Matrix ring over a semi-simple algebra

Question

I'm working on the following question and I'm a little stuck. Would appreciate any hints or solutions to any parts.

Let $A$ be a finite-dimensional semi-simple algebra over $\mathbb{C}$ and set $M_n(A)$ to be the set of $n\times n$ matrices over $A$.

1. Show that $M_2(A)$ is semisimple
2. If $dim_{\mathbb{C}}(A)$ is prime, show that $M_2(A)$ is not simple
3. If $A$ is not commutative, there is a $t\in M_2(A)$ with $t^3\neq 0$ and $t^4=0$

Here are my thoughts. For 1, we can first decompose $A$ into its product of simple pieces. And if a module is simple over a field, is it isomorphic to the field itself? So we can use this to get $M_2(A) = \oplus M_2(A_i)$ Where the $A_i$'s are the simple pieces. I feel like I'm close at this point, but how do I finish?

for part 2. I'm not sure how dim is defined here for part 3. This seems to have something to do with the fact that $M_2(A)$ has dimension 4. I'm thinking something about annihilators and order of a module would be relevant here.

Old Qualifying Exam Spring 1992