Prove $(A^T)^{-1}$ = $(A^{-1})^T$ whenever $A$ is invertible. Prove $(A^T)^{-1}$ = $(A^{-1})^T$ for any invertible matrix $A.$
I actually don't know where to start. I do not think I can just apply index laws.
Any help is cool! Thanks.
Try taking the transpose of the equation
$$AA^{-1}=I.$$
$A^T(A^T)^{-1} = I$   ;                        $ eq^n 1$
taking   transpose  on  both   side of $ eq^n 1 $
$[(A^T)^{-1}]^T A = I $;    $eq^n 1.1$
[Since $(AB)^T= B^TA^T $ &amp; $I^T=I$ , $(A^T)^T=A$]
Now  to hold above $eq^n 1.1$ it must satisfy the relation below
$[(A^T)^{-1}]^T  =  A^{-1} $   ;           $eq^n 2 $
taking transpose on $eq^n 2 , $ we get
$ (A^T)^{-1} = (A^{-1})^T $