Solution
= Solution
With the two <diagonal matrices> from the preceding part, write $D_A=PAP^{-1}$ and $D_B=PBP^{-1}$. Their diagonal products agree entry by entry, so $D_AD_B=D_BD_A$. Therefore
$$
AB=P^{-1}D_AD_BP=P^{-1}D_BD_AP=BA.
$$
Hence <simultaneous diagonalization> implies that $A,B$ are \b[<commuting operators>].