Exercise 2.5.11. Given the incidence matrix from exercise 2.5.10 with the final column removed and a diagonal matrix with elements 1, 2, 2, and 1, write the equations for the system
Show that eliminating gives the system and solve the system for . If represents the currents entering nodes 1, 2, and 3 of the graph, calculate the potentials at each node and the currents on each edge.
Answer: Removing the final column from the matrix in exercise 2.5.10 gives us the new matrix
The diagonal matrix
has as its inverse
(Recall from Note 4 of section 1.6, “Inverses and Transposes”, that the inverse of a diagonal matrix with nonzero entries on the diagonal is itself a diagonal matrix, with the entries on the diagonal of equal to the reciprocals of the corresponding entries on the diagonal of .)
We can thus express the system in matrix form as
We can express the system in matrix form as
This corresponds to the following system of equations:
Going back to the original two equations
we can eliminate by multiplying the first equation by to get or and then subtracting the second equation from this to get .
If the system can therefore be expressed in matrix form as
or as the system of equations
We begin elimination by multiplying the first equation by -1/3 and subtracting it from the second equation, and multiplying the first equation by -2/3 and subtracting it from the third equation. The result is the following system:
We then multiply the second equation by -1/4 and subtract it from the third equation:
Solving for we have . Substituting the value of into the second equation we have or so that . Finally we substitute the values of and into the first equation to obtain or so that .
The solution to the system when is thus . If represents the currents into each of nodes 1, 2, and 3 respectively then represents the potentials at each of those nodes.
What are the currents along the edges? To determine that we must solve for . Since we have or
We can check this answer using the equation
So represents the currents flowing along edges 1, 2, 3, and 4 respectively.
NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.
If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition, Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books.