Math 369 HW #9
Due at the beginning of class on Friday, November 7
Reading: Sections DM, SD.
Problems:
1. For any n × n matrix M , the characteristic polynomial of M is the polynomial
pA (x) = (−1)n det(M − x1n ).
Find both the minimum and characteristic polynomials of each of the following matrices.
Factor each polynomial.


5 0 0
(a) A = 0 5 0
0 0 5


4 0 0
(b) B = 0 5 0
0 0 6
1 π
(c) C =
0 1
√ 1
17
(d) D =
0
1
2. Answer each of the following true/false questions, giving specific support for your answer.
(Hint: Your answers to Problem 1 may come in handy.)
(a) The minimal polynomial of a matrix is the same as the characteristic polynomial.
(b) The minimal polynomial of a matrix is sometimes the same as the characteristic polynomial.
(c) Two different matrices can have the same minimal polynomial and the same characteristic polynomial.
3. Diagonalize the matrix


1 1 1
A = 1 1 1 .
1 1 1
In other words, find an invertible matrix P and a diagonal matrix D so that A = P DP −1 .
4. (a) Compute the determinant of the matrix


1 2 3
4 5 6 
7 8 10
by hand.
1
(b) Use Sage to compute the determinant of the matrix

















1 2 3 4 5 6 7 8 9 10
10 1 2 3 4 5 6 7 8 9 

9 10 1 2 3 4 5 6 7 8 

8 9 10 1 2 3 4 5 6 7 

7 8 9 10 1 2 3 4 5 6 

6 7 8 9 10 1 2 3 4 5 

5 6 7 8 9 10 1 2 3 4 

4 5 6 7 8 9 10 1 2 3 

3 4 5 6 7 8 9 10 1Building
2 
2 3 4 5 6 7 8 9 10 1
Matrices
Building Matrices
Clayton Shonkwiler
See the Sage demonstration at the end of section DM in the textbook for help.
31,toClayton
2014
Shonkwiler
(Rather than typing in 100 numbers by hand, you might think October
about how
build the
above matrix programmatically, either with your favorite scripting language or in Sage.
October 31, 2014
For example, the Sage code
nums = range (1 ,6)
A =[]
for i in range
:
nums(5)
= range
(1 ,6)
A . append
( nums [ i :]+ nums [: i ])
A =[]
B = matrix ( A )for i in range (5) :
A . append ( nums [ i :]+ nums [: i ])
will cause B to be the matrix
B
matrix ( A )
[1 2 3 4 5] [1 2 3 4 5]
[2 3 4 5 1] [2 3 4 5 1]
[3 4 5 1 2] [3 4 5 1 2]
[4 5 1 2 3] [4 5 1 2 3]
[5 1 2 3 4] [5 1 2 3 4]
You can build the 10 ×B .10determinant
matrix from()
this problem in a similar way. Sage is built
1875
on Python and the expression nums[i:] is an example of Python’s slicing syntax. See
https://docs.python.org/3.4/tutorial/introduction.html#lists for more.)
2