Page 147 - 35Linear Algebra

7.4 Review Problems 147

2 1 2 1 2 1 2 1 2 1
   
 
  x 0 2 1 2 1 0 1 2 1 2
2 1 1    
   

x y z  1 2 1  y , 0 1 2 1 2 0 2 1 2 1 ,
 

 

1 1 2 z    
0 2 1 2 1 0 1 2 1 2
0 0 0 0 2 0 0 0 0 1
 4 1   2 2   
−2 − 4 − 1 2 1
3 3 3 3
 2 − 5 3 2   5 −  4 5 2 .
  6


2  
3
3
3
−1 2 −1 12 − 16 10 7 8 2
3 3
T
T
T
2. Let’s prove the theorem (MN) = N M .
Note: the following is a common technique for proving matrix identities.
i
i
(a) Let M = (m ) and let N = (n ). Write out a few of the entries of
j
j
each matrix in the form given at the beginning of section 7.3.
(b) Multiply out MN and write out a few of its entries in the same
form as in part (a). In terms of the entries of M and the entries
of N, what is the entry in row i and column j of MN?
T
(c) Take the transpose (MN) and write out a few of its entries in
the same form as in part (a). In terms of the entries of M and the
T
entries of N, what is the entry in row i and column j of (MN) ?
(d) Take the transposes N T and M T and write out a few of their
entries in the same form as in part (a).
T
T
(e) Multiply out N M and write out a few of its entries in the same
form as in part a. In terms of the entries of M and the entries of
T
T
N, what is the entry in row i and column j of N M ?
(f) Show that the answers you got in parts (c) and (e) are the same.

1 2 0
T
T
3. (a) Let A = . Find AA and A A and their traces.
3 −1 4
T
(b) Let M be any m × n matrix. Show that M M and MM T are
symmetric. (Hint: use the result of the previous problem.) What
are their sizes? What is the relationship between their traces?

147