We can use that post again here!
The construction there uses an $R-S$ bimodule to create the ring: $T=\begin{pmatrix} R & M \\ 0 & S \\ \end{pmatrix}$ with formal matrix multiplication as the product. We can use the same thing here, except we can use an $S-R$ bimodule in the lower left corner: $T=\begin{pmatrix} R & 0 \\ M & S \\ \end{pmatrix}$. In your case, $R=M=M_2(\mathbb{C})$ and $S$ is the subring of lower triangular matrices of $R$.
Applying (analogous) info from "that post", we have $rad(T)=\begin{pmatrix} rad(R) & 0 \\ M & rad(S) \\ \end{pmatrix}$.
Can you take it from here?
Extra tools to keep in mind when doing things like this:
- if $I$ is a nilpotent (one or two sided) ideal, then $I \subseteq rad(R)$.
- if $R/I$ is semisimple, then $rad(R)\subseteq I$.
Check and see what you get when you compute $T/rad(T)$ for your example! :)