Square of transpose of two matrices

[V0ODO0CH1LD]

Homework Statement

Let A and B be two square matrices of order n such that AB = A and BA = B. Then, what is the value of [(A + B)^t]²?

Homework Equations

The Attempt at a Solution

[(A + B)^t]² = A^tA^t + A^tB^t + B^tA^t + B^tB^t.

I tried to use the fact that AB = A and BA = B to keep going but I didn't succeed at it..

 

[Yuu Suzumi]

Do you remember what $(AB)^T$ is in terms of $A^T$ and $B^T$? Have you tried using that?

 

[V0ODO0CH1LD]

yeah, the expression would look like $A^t A^t + (BA)^t + (AB)^t + B^t B^t = A^t A^t + B^t + A^t + B^t B^t$. But I have a tendency to go down paths that take me nowhere when messing around with equations and expressions.. That is where I got stuck in the first place..

 

[haruspex]

$A^t A^t + B^t + A^t + B^t B^t$

I don't understand how you would know when you have the desired answer. But you can simplify the above further. Investigate ABA.

 

[V0ODO0CH1LD]

The possible answers are:

(i) $(A+B)^2$; (ii) $2(A^t B^t)$; (iii) $2(A^t+B^t)$; (iv) $A^t+B^t$; (v) $A^tB^t$;

 

[Yuu Suzumi]

With the hint given by haruspex you can arrive at one of those!

Find two ways to write $ABA$!

 

[V0ODO0CH1LD]

$(AA)^t + B^t + A^t + (BB)^t = (ABA)^t + B^t + A^t + (BAB)^t = (AB)^t + B^t + A^t + (BA)^t = (A)^t + B^t + A^t + (B)^t = 2(A^t + B^t)$

Is that correct!?

 

[Yuu Suzumi]

Yes. At least that's what I got too :-)

Do you know where the thread is in which they describe how much we are allowed to help I am HW forums? I can't find it and therefore I am limiting myself to remarks like above.

 

[V0ODO0CH1LD]

I didn't even know there were rules on that, oh well.. Thanks!

 

[Dick]

Yes. At least that's what I got too :-)

Do you know where the thread is in which they describe how much we are allowed to help I am HW forums? I can't find it and therefore I am limiting myself to remarks like above.

https://www.physicsforums.com/showthread.php?t=414380 Solving the problem for them is right out. I'd say give them the smallest clue you think will get them heading in the right direction. The hints you gave so far worked fine, I wouldn't do more even if it is tempting to do so.

 

[Yuu Suzumi]

I'd say give them the smallest clue you think will get them heading in the right direction.

Thanks! That is a policy I can endorse. We want the OP to feel proud of his solution, after all.