I am really struggling with these type of proofs. Could someone please give me hints on how to prove them, I do know the basic properties of transpose and inverse.
If $ \mathbf{A} $ is invertible and $ \mathbf{AA}^T = \mathbf{A}^T\mathbf{A} $, show that $ (\mathbf{A}^T \mathbf{A}^{-1})^T = (\mathbf{A}^T\mathbf{A}^{-1})^{-1} $
Any help will be greatly appreciated.
On
I'll assume that you can show $(XY)^T=Y^TX^T$ and for invertible $X,Y$ we have $(XY)^{-1}=Y^{-1}X^{-1}$. (You can show these by direct verification.)
For our problem at hand, first note that $\det(A^T)=\det A$ so $A^{T}$ is invertible because $A$ is. Next, $$ (A^TA^{-1})^T=(A^TA^{-1})^{-1}\iff(A^{-1})^TA=A(A^T)^{-1}\iff(A^{-1})^TAA^T=A. $$ For the third $\iff$ above, we have right-multiplied both sides of the middle equality by $A^T$. Now the claim follows because the last equality above is true: $$ (A^{-1})^TAA^T=(A^{-1})^TA^TA=(AA^{-1})^TA=A. $$
$(A^T A^{-1})^T=(A^{-1})^T A=(A^T)^{-1}(A^{-1})^{-1}=(A^{-1}A^T)^{-1}=(A^TA^{-1})^{-1}$