Square of transpose of two matrices

  • #1
V0ODO0CH1LD
278
0

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 = AtAt + AtBt + BtAt + BtBt.

I tried to use the fact that AB = A and BA = B to keep going but I didn't succeed at it..
 
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  • #2
Do you remember what [itex] (AB)^T[/itex] is in terms of [itex]A^T[/itex] and [itex]B^T[/itex]? Have you tried using that?
 
  • #3
yeah, the expression would look like [itex] A^t A^t + (BA)^t + (AB)^t + B^t B^t = A^t A^t + B^t + A^t + B^t B^t [/itex]. 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..
 
  • #4
V0ODO0CH1LD said:
[itex]A^t A^t + B^t + A^t + B^t B^t [/itex]
I don't understand how you would know when you have the desired answer. But you can simplify the above further. Investigate ABA.
 
  • #5
The possible answers are:

(i) [itex] (A+B)^2 [/itex]; (ii) [itex] 2(A^t B^t) [/itex]; (iii) [itex] 2(A^t+B^t) [/itex]; (iv) [itex] A^t+B^t [/itex]; (v) [itex] A^tB^t [/itex];
 
  • #6
With the hint given by haruspex you can arrive at one of those!

Find two ways to write [itex]ABA[/itex]!
 
  • #7
[itex] (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) [/itex]

Is that correct!?
 
  • #8
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.
 
  • #9
I didn't even know there were rules on that, oh well.. Thanks!
 
  • #10
Yuu Suzumi said:
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.
 
  • #11
Dick said:
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.
 

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