Eigenvectors of transpose matrix, incorrect algebraic manipulation

Question

I am missing a trivial detail here which makes the following algebraic manipulation wrong but I don't know which is it:

\begin{align*} Ax &= \lambda x\\ (Ax)^T &= (\lambda x)^T \\ x^TA^T &= \lambda x^T \\ x^TA^Tx &= \lambda x^Tx \\ x^T(A^Tx) &= x^T (\lambda x) \\ A^Tx &= \lambda x. \end{align*}

I can easily can come up with counter-examples showing that the eigenvectors of the transpose are not the same but what is it exactly that makes the above algebra incorrect?

Answer 1

You can't cancel $x^T$ simply from both sides as it is not a square matrix and hence its inverse doesn't exist.

This means that there's no $(X^T){^{-1}}$ such that $X^T(X^T)^{-1} = 1$. So the last line may not be always true.