Note that
2 3
1
2
3
AB
4 5
4
2
5
2 1


−

 

= 
 

−
 



 =  
2
8
6
3 10
3
0
4
8
8
10
12
10
5
10
3
−
+
−
+
−




=




− +
+
−
+
+




and
2 3
2 12
4
6
6
15
1
2
3
BA
4 5
4
20
8 10
12
25
4
2
5
2 1
2
4
4
2
6
5
−
− +
+




−






=
=
−
− +
+






−





−
− +
+




 
10
2
21
16
2
37
2
2
11
−



= −




−
−


Clearly AB ≠ BA
In the above example both AB and BA are of different order and so AB ≠ BA But
one may think that perhaps AB and BA could be the same if they were of the same
order But it is not so, here we give an example to show that even if AB and BA are of
same order they may not be same Example 14 If 
1
0
A
0
1


= 
−


 and 
0
1
B
1
0


= 



, then 
0
1
AB
1
0


= 

−
