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?

Try e.g. $$M = \pmatrix{3 & 2\cr -1 & 0\cr}$$
Eigenvalues are $1$ and $2$.  Normalized right eigenvectors form the matrix
$$A = \pmatrix{-1/\sqrt{2} & -2/\sqrt{5} \cr 1/\sqrt{2} & 1/\sqrt{5}\cr}$$
Normalized left eigenvectors form
$$ B = \pmatrix{1/\sqrt{5} & 2/\sqrt{5}\cr 1/\sqrt{2} & 1/\sqrt{2}\cr}$$
These are not inverses.
