The definition of a graph's transitive reduction
If we order the vertices alphabetically, then the adjacency matrix is
$$A=\begin{pmatrix}
0&amp;1&amp;1&amp;1&amp;0\\
0&amp;0&amp;0&amp;1&amp;0\\
0&amp;0&amp;0&amp;1&amp;1\\
0&amp;0&amp;0&amp;0&amp;1\\
0&amp;0&amp;0&amp;0&amp;0
\end{pmatrix}$$
The adjacency matrix of the transitive closure would be
$$B=\begin{pmatrix}
0&amp;1&amp;1&amp;1&amp;1\\
0&amp;0&amp;0&amp;1&amp;1\\
0&amp;0&amp;0&amp;1&amp;1\\
0&amp;0&amp;0&amp;0&amp;1\\
0&amp;0&amp;0&amp;0&amp;0
\end{pmatrix}$$
To get the this one you get each row by navigate the graph starting at the node corresponding to that row and mark with a $1$ the columns of the nodes that you manage to visit and with $0$ the rest.
Now,
$$AB=\begin{pmatrix}
0&amp;0&amp;0&amp;2&amp;3\\
0&amp;0&amp;0&amp;0&amp;1\\
0&amp;0&amp;0&amp;0&amp;1\\
0&amp;0&amp;0&amp;0&amp;0\\
0&amp;0&amp;0&amp;0&amp;0
\end{pmatrix}$$
To build the adjacency matrix $C$ of the transitive reduction put a $1$ in position $(i,j)$ if the $(i,j)$ entry of $A$ is non-zero and the $(i,j)$ entry of $AB$ is zero.
So, the adjacency matrix of the transitive reduction should be
$$C=\begin{pmatrix}
0&amp;1&amp;1&amp;0&amp;0\\
0&amp;0&amp;0&amp;1&amp;0\\
0&amp;0&amp;0&amp;1&amp;0\\
0&amp;0&amp;0&amp;0&amp;1\\
0&amp;0&amp;0&amp;0&amp;0
\end{pmatrix}$$
So, the transitive reduction looks like this