Page 146 - 35Linear Algebra

146 Matrices

Example 92 Continuing from the previous example,

1 1 1 0
M = , N = .
0 1 1 1
so

2 1 1 1
MN = 6= NM = .
1 1 1 2
However, tr(MN) = 2 + 1 = 3 = 1 + 2 = tr(NM).

Another useful property of the trace is that:

tr M = tr M T

This is true because the trace only uses the diagonal entries, which are fixed
by the transpose. For example,

T
1 1 1 2 1 2
tr = 4 = tr = tr .
2 3 1 3 1 3
Finally, trace is a linear transformation from matrices to the real numbers.
This is easy to check.

7.4 Review Problems

Webwork: Reading Problems 2 , 3 , 4

1. Compute the following matrix products

1
 
   4 1 
2
1 2 1 −2 −  
3 3  

 ,
4 5 2  2 − 5 3 2  1 2 3 4 5 3 ,
 

 
3
 
7 8 2 −1 2 −1 4
5
 
1
     
2
  1 2 1 −2 4 − 1 1 2 1
3 3
 
5
3 1 2 3 4 5 , 4 5 2  2 − 3 2   
 4 5 2 ,
 
 

3
 
7 8 2 −1 2 −1 7 8 2
4
5
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