Left/Right Eigenvectors Let $M$ be a nonsymmetric matrix; suppose the columns of matrix $A$ are the right eigenvectors of $M$ and the rows of matrix $B$ are the left eigenvectors of $M$. 
In one of the answers to a question on left and right eigenvectors it was claimed that $AB=I$. Is that true, and how would you prove it?

Here's a tentative proof that $A$ is the inverse of $B$. 
The left eigenvectors of $M$ are the right eigenvectors of $M^T$. Therefore if $M=B⋅Λ⋅B^{-1}$ then $M^T=B^{-T}⋅Λ⋅B^T$ which is an EVD. If the eigenvalues are distinct, the EVD is unique (up to scaling of eigenvectors), which proves that left eigenvectors are rows of $B^{−1}$, i.e. $A=B^{-1}$. 
(I'm pasting this answer from Power of a Nonsymmetric Matrix , it's due to Florian)
Of course the counterexamples below cast a dark shadow on this proof, but I can see nothing wrong with it.
