Page 53 - Matrix Analysis & Applied Linear Algebra

46 Chapter 2 Rectangular Systems and Echelon Forms
Exercises for section 2.1

2.1.1. Reduce each of the following matrices to row echelon form, determine
the rank, and identify the basic columns.
211 30 41
 
123
 
   424 41
1233 55 
 268   213 10 

(a)  2469  (b)  260  (c)  81 43 


 634

2676  125  95 
 003 −30 03 
386
842 14 1133
2.1.2. Determine which of the following matrices are in row echelon form:
   
123 0000
(a)  004  . (b)  0100  .
010 0001
 
  120010
223 −4
(c)  007 −8  . (d)  000001  .
 000100 
000 −1
000000
2.1.3. Suppose that A is an m × n matrix. Give a short explanation of why
each of the following statements is true.
(a) rank (A) ≤ min{m, n}.
(b) rank (A) <m if one row in A is entirely zero.
(c) rank (A) <m if one row in A is a multiple of another row.
(d) rank (A) <m if one row in A is a combination of other rows.
(e) rank (A) <n if one column in A is entirely zero.
 
.1 .2 .3
2.1.4. Let A =  .4 .5 .6  .
.7 .8 .901
(a) Use exact arithmetic to determine rank (A).
(b) Now use 3-digit floating-point arithmetic (without partial piv-
oting or scaling) to determine rank (A). This number might be
called the “3-digit numerical rank.”
(c) What happens if partial pivoting is incorporated?

2.1.5. How many different “forms” are possible for a 3 × 4 matrix that is in
row echelon form?

2.1.6. Suppose that [A|b] is reduced to a matrix [E|c].
(a) Is [E|c] in row echelon form if E is?
(b) If [E|c] is in row echelon form, must E be in row echelon form?