Proving Commutativity of Matrices: A + B and (A+B)^2 Equations Validity

[Physicsissuef]

Homework Statement

Prove that if the matrices A and B commute (AB=BA) than the equations:

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

are valid.

Homework Equations

The Attempt at a Solution

How will I start solving this task? Should I chose two commute matrices ? How will I know what matrices to choose?

 

[Defennder]

You don't have to pick any two arbitrary commutative matrices. In fact if you did, it would have been wrong because you are asked to prove it, not verify it.

For a start, note that in matrix multiplication, the order usually matters, AB is not the same as BA. Let C = A+B. Now you have (A+B)^2 = C(A+B). Multiply the matrix on the left to the one on the right component wise, you'll get CA + CB. Replace C with A+B, then multiply it on the right. Then make use of the fact that AB = BA to get the above.

 

[Physicsissuef]

And can I solve it like this:

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

AB=BA

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

?

 

[HallsofIvy]

Yes, of course!

There is a typo in your second question: it obviously should be (A+ B)³.

Now that you know that (A+ B)²= A²+ 2AB+ B², multiply again!

 

[Physicsissuef]

Ok. Thanks. I have another (I believe easy), but I can't figure out.

Choose two quadric matrices A and B (matrices with same number of rows and columns):

a)2 rows, 2 columns ; b)3 rows, 3 columns

and check the identity det(AB)=detA * detB

I know how to find detA and detB but what is detAB?

 

[Defennder]

Uh what do you mean by what is detAB? det(AB) is the determinant of the matrix obtained by multiplying matrix A to B on the left.

 

[Physicsissuef]

Oh, I think I know what is the point.
det(AB), I should first find A*B and then find the determinant.

det(A) I should find the determinant of A

det(B) I should find the determinant of A

And then check det(AB)=det(A)det(B)