Chapter 7.4, Problem 69E

We say that two n x n matrices A and B are simultaneously diagonalizable if there exists an invertible $n\times n$ matrix S such that $S^{-1}AS$ and $S^{-1}BS$ are both diagonal.
a. Are the matrices
$A=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$ and $B=\left[\begin{array}{ccc}1& 2& 3\\ 0& 2& 3\\ 0& 0& 3\end{array}\right]$
simultaneously diagonalizable? Explain.
b. Show that if A andB are simultaneously diagonalizable then $AB=BA$ .
c. Give an example of two $n\times n$ matrices such that $AB=BA$ , but A and B are not simultaneously diagonalizable.
d. Let D be a diagonal $n\times n$ matrix with n distinct entries on the diagonal. Find all $n\times n$ matrices B that commute with D.
e. Show that if $AB=BA$ and A has n distinct eigenvalues, then A and B are simultaneously diagonalizable. Hint: Part d is useful.