Can I do the same things on both sides when I have a not equal sign? I am trying to prove this statement:

If $A$ is not symmetric then $A^{-1}$ is not symmetric 

And the following is my proof
If $A≠A^T$,
\begin{align*}
& A^{-1}A=I≠A^{-1} A^T \\
& I^T \neq \left(A^{-1} A^T \right)^T \\
& I \neq A \left( A^{-1} \right)^T \\
& A^{-1} \neq \left( A^{-1} \right)^T \\
& A^{-1} \neq \left( A^{-1} \right)^T. \\
\end{align*}
I am doing several operations on both sides which usually is justified when there is a equal sign between RHS and LHS. I am not sure if this is valid when there is NOT equal sign. 

That way is fine because you are using not equality which is an inequality and when you deal with these it is still fundamental to do to both sides of the equation what you do to one side of the equation
