I have to determine the Jacobson radical of this matrix ring: $\begin{pmatrix} \mathbb{Z}_{63} & \mathbb{Z}_{63}\\ 0& \mathbb{Z}_{63} \end{pmatrix}$
I have done the following $(a,b,c,r,s,t \in \mathbb{Z}_{63})$:
$$\begin{pmatrix} 1 & 0\\ 0& 1 \end{pmatrix}- \begin{pmatrix} r & s\\ 0& t \end{pmatrix}\begin{pmatrix} a & b\\ 0& c \end{pmatrix}= \begin{pmatrix} 1-ra & -(rb+sc)\\ 0& 1-tc \end{pmatrix}$$
Then I have: $$\begin{pmatrix} 1-ra & -(rb+sc)\\ 0& 1-tc \end{pmatrix}^{\!\!-1}= \frac{1}{1-tc-ra+ratc}\begin{pmatrix} 1-tc & rb + sc\\ 0 & 1-ra \end{pmatrix}.$$
What should be my next step?
It's quite a bit easier to deduce the elements in another way.
First of all, notice that $I=\begin{bmatrix}0&\mathbb {Z}_{63}\\0&0\end{bmatrix}$ is a nilpotent ideal, so it is contained in the Jacobson radical. Finding the maximal ideals of the ring is therefore equivalent to finding the maximal ideals of $R/I\cong \mathbb Z_{63}\times \mathbb Z_{63}$.
But this is easy, right? The maximal ideals in $\mathbb Z_{63}$ are $(3)$ and $(7)$, so the maximal ideals of the product are $(3)\times \mathbb Z_{63}$, $(7)\times \mathbb Z_{63}$ and $\mathbb Z_{63}\times (3)$ and $\mathbb Z_{63}\times (7)$. This makes the radical of the product ring $\mathbb (21)\times (21)$.
Lifting this back up to the original ring, you have
$$ \begin{bmatrix} (21)&\mathbb Z_{63}\\0&(21)\end{bmatrix} $$
You can find similar descriptions of what's going on here:
https://math.stackexchange.com/a/1050265/29335