Page 59 - Matrix Analysis & Applied Linear Algebra

52 Chapter 2 Rectangular Systems and Echelon Forms

2.2.2. Construct a matrix A whose reduced row echelon form is

120 −3000
 
 001 −4010 
 000 0100 


 .
 000 0001 
E A = 
000 0000
 
000 0000
Is A unique?
2.2.3. Suppose that A is an m × n matrix. Give a short explanation of why
rank (A) <n whenever one column in A is a combination of other
columns in A .

2.2.4. Consider the following matrix:

 
.1 .2 .3
A =  .4 .5 .6  .
.7 .8 .901

(a) Use exact arithmetic to determine E A .
(b) Now use 3-digit floating-point arithmetic (without partial piv-
oting or scaling) to determine E A and formulate a statement
concerning “near relationships” between the columns of A .

2.2.5. Consider the matrix
 
10 −1
E =  01 2  .
00 0
You already know that E ∗3 can be expressed in terms of E ∗1 and E ∗2 .
However, this is not the only way to represent the column dependencies
in E . Show how to write E ∗1 in terms of E ∗2 and E ∗3 and then
express E ∗2 as a combination of E ∗1 and E ∗3 . Note: This exercise
illustrates that the set of pivotal columns is not the only set that can
play the role of “basic columns.” Taking the basic columns to be the
ones containing the pivots is a matter of convenience because everything
becomes automatic that way.