# Homework Help: Matrix proves

1. May 28, 2008

### Physicsissuef

1. The problem statement, all variables and given/known data

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.

2. Relevant equations

3. 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?

2. May 28, 2008

### 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.

3. May 28, 2008

### Physicsissuef

And can I solve it like this:

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

AB=BA

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

?

4. May 28, 2008

### 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!

5. May 28, 2008

### 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?

6. May 28, 2008

### 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.

7. May 28, 2008

### 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)