Page 83 - Matrix Analysis & Applied Linear Algebra

76 Chapter 2 Rectangular Systems and Echelon Forms

The point of this discussion is to conclude that the more general 11 × 6
rectangular system can be replaced by an equivalent 6 × 6 square system that
has a unique solution by dropping the last nodal equation and using only the
simple loop equations. This is characteristic of practical work in general. The
physics of a problem together with natural constraints can usually be employed
to replace a general rectangular system with one that is square and possesses a
unique solution.
One of the goals in our study is to understand more clearly the notion of
“independence” that emerged in this application. So far, independence has been
an intuitive idea, but this example helps make it clear that independence is a
fundamentally important concept that deserves to be nailed down more firmly.
This is done in §4.3, and the general theory for obtaining independent equations
from electrical circuits is developed in Examples 4.4.6 and 4.4.7.

Exercises for section 2.6

2.6.1. Suppose that R i = i ohms and E i = i volts in the circuit shown in
Figure 2.6.1.
(a) Determine the six indicated currents.
(b) Select node number 1 to use as a reference point and fix its
potential to be 0 volts. With respect to this reference, calculate
the potentials at the other three nodes. Check your answer by
verifying the loop rule for each loop in the circuit.

2.6.2. Determine the three currents indicated in the following circuit.

5Ω 8Ω
I 2 I 1
1Ω 1Ω
12 volts 9 volts
10Ω
I 3

2.6.3. Determine the two unknown EMFs in the following circuit.
20 volts
6Ω
1 amp
E 1
4Ω
2 amps
E 2 2Ω