prove that the inverse of an invertible symmertric matrix is symmetric.

Hi madukar!!

Note about notation:
If A is a square matrix, then we will note with (A)^t the transpose of
A.
If A is an inversible square matrix, then we will note with (A)^-1 the
inverse of A.
I is the identity matrix.

Definition: a square matrix A is a symmetric matrix if and only if A =
(A)^t

Property: (A.B)^t = (A)^t . (B)^t 

Proposition: the inverse of an invertible symmetric matrix is
symmetric.
Proof:
A = (A)^t , then 
I = (A)^-1 . A   (eq.1)

I = (I)^t = ((A)^-1 . A)^t = 
          = ((A)^-1)^t . (A)^t =   (because A is symmetric)
          = ((A)^-1)^t . A         (eq.2)

Then by eq.1 and eq.2 we have:

(A)^-1 . A = ((A)^-1)^t . A   

then multiplying both sides of the equation by (A)^-1 we have:

(A)^-1 . A . (A)^-1= ((A)^-1)^t . A . (A)^-1  

then like A . (A)^-1 = I we have:

(A)^-1 = ((A)^-1)^t  

That prove the proposition.

I did it by my own knowledge, if you have doubts, please post a
request of clarification.
Regards.
livioflores-ga

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Request for Answer Clarification by madukar-ga on 08 Dec 2002 13:58 PST

hi, 

  what does (A)^ -1 mean by? is it A inverse?

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Clarification of Answer by livioflores-ga on 08 Dec 2002 14:06 PST

Yes madukar, I write it at the start with the notes about notation. We
must be very carefully with this thing in math answers becaeuse we are
very limited writting mathematics symbols.

Feel free to request more clarifications if they are needed.
livioflores-ga

The Property: 

" (A.B)^t = (A)^t . (B)^t "

should be 

(A.B)^t = (B)^t . (A)^t   (noticed the swapped order here)

This holds for any two matrices for which multiplication is defined, 
not just square ones.

To get (eq.2) you just need to change

I = (I)^t = ((A)^-1 . A)^t = ...

to 

I = (I)^t = (A . (A)^-1)^t = ...
          
The rest of the proof is exactly the same.

Yes!! You īre right eldog.

It was a typo at start of the demonstration...then I continue without noticing it.

Thank you.