4. (CS application: graph theory) This problem is based on an example in [HPOS] Figure 2 shows a network in which the nodes A, B, C, and E are input nodes and node D an output node. For example, some type of liquid may enter the network at nodes A, B, C, or E but ultimately leaves it at node D. There is a lot of redundancy in the network and we would like to eliminate the redundant ares while stl ensuring that anything entering one of the input nodes can ultimately reach the output node. In the language of graph theory, we are producing a spanning tree. To do this: (a) Set up the system of equations representing the network flow of the nodes and represent it as a matrix; this will be the A in AF-. To do so, for each node the sum of the input Pigure 2: Network for problem4 values should equal the sum of the output values. For example, the equation for node A Note that the system of equations will consist of six variables but only five equations. (b) Find the reduced row echelon form (RREF) of the matrix. (c) Eliminate from the network the ares corresponding to the free variables; that is, keep the arcs corresponding to the pivot variables. Draw the final network with the redundant ares eliminated. Note that the directions of the ares matters when constructing the system of equations, but don't matter in the final network since we assume that whatever is flowing through the network can flow in either direction of an aro