Proof: Product of Orthogonal Matrices is Orthogonal

[soothsayer]

Homework Statement

Show that a product of orthogonal matrices is orthogonal.

Homework Equations

Orthoganol matrix: M^-1=M^T

The Attempt at a Solution

since A^-1=A^T
A^-1 and A^T commute.
commutable => symmetric => A^-1A^T=(A^-1A^T)^T

(A^-1A^T)^-1=A^-1A^T/det(A^-1A^T)
=> (A^-1A^T)^-1(A^-1A^T)=(A^-1A^T)²/det(A^-1A^T)
M^-1M=I
M²=-I
=> I=-I/det(A^-1A^T)=> det(A^-1A^T) = -1
=> (A^-1A^T)^-1=-(A^-1A^T)
=> (A^-1A^T)^-1=/=A^-1A^T ??
I'm sure I did something wrong here. Probably overstepped an assumption. Can anyone help?

 

[tiny-tim]

hi soothsayer!

i don't understand what you're trying to do …

you have to prove that if A^-1=A^T and B^-1=B^T, then (AB)^-1=(AB)^T

 

[soothsayer]

Yes, I understand that, but what practical way do I have of proving that?

 

[tiny-tim]

what's another way of writing (AB)^-1 ?

 

[soothsayer]

Well, for orthoganol A and B, it would be = (AB)^T and I believe you can also write it as B^-1A^-1, correct?

 

[tiny-tim]

hi soothsayer!

(just got up :zzz: …)

what's another way of writing (AB)^-1 ?

… I believe you can also write it as B^-1A^-1, correct?

correct!

ok, now what's another way of writing (AB)^T ?