Proof $(10$ points$)$

Let A$,$BinR$^($n$\backslash$times n$).$ The eigenvalues and eigenvectors of A are given by $(\backslash$alpha $\_(1),$vec$($v$)\_(1)),(\backslash$alpha $\_(2),$vec$($v$)\_(2)),$cdots,$(\backslash$alpha $\_($n$),$vec$($v$)\_($n$)),$ where

all the $\backslash$alpha $\_($iin$,$ are distinct. Similarly the eigenvalues and eigenvectors of B are given by $(\backslash$beta $\_(1),$vec$($v$)\_(1)),$

$(\backslash$beta $\_(2),$vec$($v$)\_(2)),$cdots,$(\backslash$beta $\_($n$),$vec$($v$)\_($n$)),$ where all the $\backslash$beta $\_($iin$,$ are distinct.

NOTE: A$,$ B have identical eigenvectors.

Prove that:

ABvec$($x$)=$BAvec$($x$),$

for any vector vec$($x$)$inR$^($n$).$