[Solved] 18.06 Exercise 12-Graphs, networks, and incidence matrices

Problem 12.1: (8.2 #1. Introduction to Linear Algebra: Strang) Write down the four by four incidence matrix A for the square graph, shown below.

(Hint: the first row has -1 in column 1 and +1 in column 2.) What vectors (x₁, x₂, x₃, x₄) are in the nullspace of A? How do you know that (1,0,0,0) is not in the row space of A?

Problem 12.2: (8.2 #7.) Continuing with the network from problem one, suppose the conductance matrix is

0

C 0 .

Multiply matrices to find A^TCA. For f = (1, 0, −1, 0), find a solution to

A^TCAx = f. Write the potentials x and currents y = −CAx on the square graph (see above) for this current source f going into node 1 and out from node 3.