Page 120 - Matrix Analysis & Applied Linear Algebra

114 Chapter 3 Matrix Algebra

3.6.9. A particular electronic device consists of a collection of switching circuits
that can be either in an ON state or an OFF state. These electronic
switches are allowed to change state at regular time intervals called clock
cycles. Suppose that at the end of each clock cycle, 30% of the switches
currently in the OFF state change to ON, while 90% of those in the ON
state revert to the OFF state.
(a) Show that the device approaches an equilibrium in the sense
that the proportion of switches in each state eventually becomes
constant, and determine these equilibrium proportions.
(b) Independent of the initial proportions, about how many clock
cycles does it take for the device to become essentially stable?

3.6.10. Write the following system in the form T n×n x = b, where T is block
triangular, and then obtain the solution by solving two small systems as
described in Example 3.6.7.

x 1 + x 2 +3x 3 +4x 4 = − 1,
2x 3 +3x 4 = 3,
x 1 +2x 2 +5x 3 +6x 4 = − 2,
x 3 +2x 4 = 4.

3.6.11. Prove that each of the following statements is true for conformable ma-
trices.
(a) trace (ABC)= trace (BCA)= trace (CAB).
(b) trace (ABC) can be different from trace (BAC).
T T
(c) trace A B = trace AB .

3.6.12. Suppose that A m×n and x n×1 have real entries.
T
(a) Prove that x x = 0 if and only if x = 0.
T
(b) Prove that trace A A = 0 if and only if A = 0.