Math 33A, Spring 2025
Worksheet, Week 10
TA: Roman Krutowski
Problem 1. For each matrix A below compare its geometric and algebraic multiplicities
to determine whether a given matrix is diagonalizable. If so, diagonalize it, i.e., find a
diagonal matrix D and matrix S such that D = S −1 AS, and verify such equation by direct
computation.
2 3
(a)
;
4 5
1 1 0
(b) 0 2 2 .
0 0 3
Problem 2. For each matrix A below compute An where:
8 −1
(a) A =
, n = 5;
6 3
1 0 0
(b) A = −5 0 2 ,
0 0 1
n = 21.
Math 33A, Spring 2025
Worksheet, Week 10
TA: Roman Krutowski
Problem 3. We say that two n × n matrices A and B are simultaneously diagonalizable if
there exists an invertible n × n matrix S such that S −1 AS and S −1 BS are both diagonal.
(a) Are the matrices
1 0 0
A = 0 1 0
0 0 1
1 2 3
and B = 0 2 3
0 0 3
simultaneously diagonalizable? Explain.
(b) Show that if A and B are simultaneously diagonalizable then AB = BA.
(c) Give an example of two n × n matrices such that AB = BA, but A and B are not
simultaneously diagonalizable.
0
