Page 22 - Matrix Analysis & Applied Linear Algebra

14 Chapter 1 Linear Equations

(a) If at the end of one minute there are 12, 25, 26, and 37 insects
in chambers #1, #2, #3, and #4, respectively, determine what
the initial distribution had to be.
(b) If the initial distribution is 20, 20, 20, 40, what is the distribution
at the end of one minute?

1.2.12. Show that the three types of elementary row operations discussed on
p. 8 are not independent by showing that the interchange operation
(1.2.7) can be accomplished by a sequence of the other two types of row
operations given in (1.2.8) and (1.2.9).

1.2.13. Suppose that [A|b] is the augmented matrix associated with a linear
system. You know that performing row operations on [A|b] does not
change the solution of the system. However, no mention of column oper-
ations was ever made because column operations can alter the solution.
(a) Describe the effect on the solution of a linear system when
columns A ∗j and A ∗k are interchanged.
(b) Describe the effect when column A ∗j is replaced by αA ∗j for
α =0.
(c) Describe the effect when A ∗j is replaced by A ∗j + αA ∗k .
Hint: Experiment with a 2 × 2or3 × 3 system.

1.2.14. Consider the n × n Hilbert matrix defined by
 1 1 1 
1 ···
2 3 n
 
1 1 1 1
 
 ··· 
 2 3 4 n+1 
 
 
1 1 1 1
H =  ···  .
 3 4 5 n+2 
 . . . . 
 . . . . 
. . . ··· .
 
 
1 1 1 1
n n+1 n+2 ··· 2n−1
Express the individual entries h ij in terms of i and j.
1.2.15. Verify that the operation counts given in the text for Gaussian elimi-
nation with back substitution are correct for a general 3 × 3 system.
If you are up to the challenge, try to verify these counts for a general
n × n system.
1.2.16. Explain why a linear system can never have exactly two different solu-
tions. Extend your argument to explain the fact that if a system has more
than one solution, then it must have infinitely many different solutions.