Page 108 - Matrix Analysis & Applied Linear Algebra

102 Chapter 3 Matrix Algebra

and
11 11 11 11 16
 
 11 11 11 11 16 
3
2
4
C + C + C + C =  11 11 11 11 16  .


11 11 11 11 16
 
16 16 16 16 28
3
The fact that [C ] 12 = 3 means there are exactly 3 three-flight routes from city
4
A to city B, and [C ] 12 = 7 means there are exactly 7 four-flight routes—try
3
4
2
to identify them. Furthermore, [C + C + C + C ] 12 = 11 means there are 11
routes from city A to city B that require no more than 4 flights.
Exercises for section 3.5
     
1 −23 12 1
2
3.5.1. For A =  0 −54  , B =  04  , and C =   , compute
4 −38 37 3
the following products when possible.
2
2
T
(a) AB, (b) BA, (c) CB, (d) C B, (e) A , (f) B ,
T
T
T
T
T
(g) C C, (h) CC , (i) BB , (j) B B, (k) C AC.
3.5.2. Consider the following system of equations:
2x 1 + x 2 + x 3 = 3,
+2x 3 =10,
4x 1
= − 2.
2x 1 +2x 2
(a) Write the system as a matrix equation of the form Ax = b.
(b) Write the solution of the system as a column s and verify by
matrix multiplication that s satisfies the equation Ax = b.
(c) Write b as a linear combination of the columns in A.
 
100
3.5.3. Let E =  010  and let A be an arbitrary 3 × 3 matrix.
301
(a) Describe the rows of EA in terms of the rows of A.
(b) Describe the columns of AE in terms of the columns of A.

3.5.4. Let e j denote the j th unit column that contains a 1 in the j th
position and zeros everywhere else. For a general matrix A n×n , describe
T
(b) e A T
the following products. (a) Ae j (c) e Ae j
i i