Are Commutative Matrices the Key to Solving These Matrix Equations?

[Theofilius]

Homework Statement

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

a) $$(A+B)^2 = A^2 + 2AB + B^2$$ ; b)$$(A+B)^3=A^3+3A^2B+3AB^2+B^3$$

are true.

Homework Equations

The Attempt at a Solution

 

[cristo]

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?

 

[Theofilius]

I don't know what do you mean?

 

[cristo]

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?

 

[Theofilius]

What do you get when you multiply out (A+B)(A+B), where A and B are matrices?

I get
$$(A+B)(A+B)=A^2+AB+BA+B^2$$

 

[cristo]

I get
$$(A+B)(A+B)=A^2+AB+BA+B^2$$

Good, now can you simplify this in the case that A and B commute?

 

[Theofilius]

Good, now can you simplify this in the case that A and B commute?

$$(A+B)^2=(A+B)(A+B)=A^2+AB+BA+B^2=A^2+2AB (or 2BA) + B^2$$

like this?

 

[cristo]

$$(A+B)^2=(A+B)(A+B)=A^2+AB+BA+B^2=A^2+2AB (or 2BA) + B^2$$

like this?

Correct!

Now, try applying similar techniques to the second question.

 

[Theofilius]

$$(A+B)^3=(A+B)(A+B)(A+B)=(A^2+2AB+B^2)(A+B)=A^3+A^2B+2A^2B+2AB^2+B^2A+B^3=A^3+A*AB+2A*AB+2<br /> <br /> AB*B+B*BA+B^3$$
$$=A^3+3A^2B+3AB^2+B^3$$

Something like this?

 

[BrendanH]

*Applause*

Huzza, well done!