Thread: Proof help (lin. algebra ?)

Okay so I was practicing proofs and the following appeared in my book. i'm confused as to how do it so i would be thankful for any help:

Let A be a symmetric nonsingular matrix. Prove that its inverse (A^-1) is symmetric.

Originally Posted by buckaroobill

Okay so I was practicing proofs and the following appeared in my book. i'm confused as to how do it so i would be thankful for any help:

Let A be a symmetric nonsingular matrix. Prove that its inverse (A^-1) is symmetric.

Let * mean transpose.

We know that,
A*=A
Because it is symettric.

We need to show,
(A^-1)*=A^-1 (1)

You can use the fact that,
(M^-1)*=(M*)^-1
For non-singular matrices.

But that means in (1) we have,
(A^-1)*=(A*)^-1=A^-1
Thus, it is symmettric.