# Commutative matrices

1. 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.

2. Homework Equations

3. The Attempt at a Solution

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cristo
Staff Emeritus
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?

I don't know what do you mean?

cristo
Staff Emeritus
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?

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
Staff Emeritus
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?

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

like this?
Correct!

Now, try applying similar techniques to the second question.

$$(A+B)^3=(A+B)(A+B)(A+B)=(A^2+2AB+B^2)(A+B)=A^3+A^2B+2A^2B+2AB^2+B^2A+B^3=A^3+A*AB+2A*AB+2 AB*B+B*BA+B^3$$
$$=A^3+3A^2B+3AB^2+B^3$$

Something like this?

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*Applause*

Huzza, well done!