Are Commutative Matrices the Key to Solving These Matrix Equations?

  • Thread starter Theofilius
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In summary, the commutativity of two matrices A and B implies that the equations (A+B)^2 = A^2 + 2AB + B^2 and (A+B)^3 = A^3 + 3A^2B + 3AB^2 + B^3 are true. This can be shown by expanding out the brackets and simplifying in the case of A and B commuting.
  • #1
Theofilius
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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.

Homework Equations





The Attempt at a Solution

 
Last edited:
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  • #2
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?
 
  • #3
I don't know what do you mean?
 
  • #4
Theofilius said:
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?
 
  • #5
cristo said:
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]
 
  • #6
Theofilius said:
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?
 
  • #7
cristo said:
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?
 
  • #8
Theofilius said:
[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.
 
  • #9
[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:
  • #10
*Applause*

Huzza, well done!
 

1. What is a commutative matrix?

A commutative matrix is a square matrix in which the order of multiplication does not affect the result. This means that if two matrices are commutative, multiplying them in either order will produce the same result.

2. How do I determine if two matrices are commutative?

To determine if two matrices are commutative, you can multiply them in both orders and compare the results. If the results are the same, then the matrices are commutative. You can also check if the matrices have the same elements in the same positions.

3. Are all matrices commutative?

No, not all matrices are commutative. In order for a matrix to be commutative, it must be a square matrix and also fulfill the commutative property of multiplication, which is not always the case for matrices.

4. What is the significance of commutative matrices?

Commutative matrices have many applications in mathematics and science, particularly in linear algebra and quantum mechanics. They allow for simplification of calculations and can help identify symmetries in systems.

5. Can non-square matrices be commutative?

No, only square matrices can be commutative. This is because non-square matrices do not have the same number of rows and columns, and therefore cannot be multiplied in the same order to produce the same result.

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