• Support PF! Buy your school textbooks, materials and every day products Here!

Commutative matrices

  • Thread starter Theofilius
  • Start date
86
0
1. Homework Statement

Proof that if two matrices A and B are commutative, BA=AB, then the equations:

a) [tex](A+B)^2 = A^2 + 2AB + B^2[/tex] ; b)[tex](A+B)^3=A^3+3A^2B+3AB^2+B^3[/tex]

are true.

2. Homework Equations



3. The Attempt at a Solution
 
Last edited:

Answers and Replies

cristo
Staff Emeritus
Science Advisor
8,056
72
Have you tried anything? What do you get if you explicitly expand out the brackets in each case? What happens in the case of A and B commuting?
 
86
0
I don't know what do you mean?
 
cristo
Staff Emeritus
Science Advisor
8,056
72
I don't know what do you mean?
What do you get when you multiply out (A+B)(A+B), where A and B are matrices?
 
86
0
What do you get when you multiply out (A+B)(A+B), where A and B are matrices?
I get
[tex](A+B)(A+B)=A^2+AB+BA+B^2[/tex]
 
cristo
Staff Emeritus
Science Advisor
8,056
72
I get
[tex](A+B)(A+B)=A^2+AB+BA+B^2[/tex]
Good, now can you simplify this in the case that A and B commute?
 
86
0
Good, now can you simplify this in the case that A and B commute?

[tex](A+B)^2=(A+B)(A+B)=A^2+AB+BA+B^2=A^2+2AB (or 2BA) + B^2[/tex]

like this?
 
cristo
Staff Emeritus
Science Advisor
8,056
72
[tex](A+B)^2=(A+B)(A+B)=A^2+AB+BA+B^2=A^2+2AB (or 2BA) + B^2[/tex]

like this?
Correct!

Now, try applying similar techniques to the second question.
 
86
0
[tex](A+B)^3=(A+B)(A+B)(A+B)=(A^2+2AB+B^2)(A+B)=A^3+A^2B+2A^2B+2AB^2+B^2A+B^3=A^3+A*AB+2A*AB+2

AB*B+B*BA+B^3[/tex]
[tex]=A^3+3A^2B+3AB^2+B^3[/tex]

Something like this?
 
Last edited:
63
0
*Applause*

Huzza, well done!
 

Related Threads for: Commutative matrices

Replies
6
Views
5K
  • Last Post
Replies
13
Views
13K
  • Last Post
Replies
13
Views
2K
  • Last Post
Replies
2
Views
6K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
3
Views
970
  • Last Post
Replies
3
Views
1K
Top