Mathematics 369
Homework (due Mar 22)
A. Hulpke
39) For each of the following matrices, determine (with explanation) whether the matrix
can be diagonalized. (You do not need to diagonalize it!) To save you calculations, the
characteristic
polynomial is given ineach case:

2
0
0 0
0
 −16 −72 −316 0

97


3
2

950 0 −291 
A =  48 222
, χA (x) = (x − 2) (x − 3)
 10
36
147 3 −45 
144 666 2844 0 −871


2
0
0 0 0
 14 −3 −5 0 0 


3
2
5
7 0 0 
B=
 −14
, χB (x) = (x − 2) (x − 3)
 −28 10
8 3 0 
0
0
0 0 3


341 −90 0 −48
30
 1348 −356 0 −192 120 


, χC (x) = (x − 1)(x − 2)(x − 3)(x − 4)(x − 5)
−50
12
3
−4
7
C=


 −112
30 0
21 −10 
46 −12 0
−4
6
40∗ ) The trace tr(A) of an n × n matrix A is the sum of its diagonal entries: tr(A) = a11 +
a22 + · · · + ann .
a) Show that for n × n matrices A, B we have that tr(AB) = tr(BA).
(Hint: Use the formula ∑ j ai j b jk for matrix multiplication.)
b) Show that similar matrices have the same trace.
41)
The Fibonacci numbers Fn are defined by the recurrence
F0 = F1 = 1,
Fn = Fn−1 + Fn−2 (n ≥ 2).
We want to determine a nonrecursive formula for the Fibonacci numbers. For this we will
express them using matrix arithmetic.
a) For n ≥ 1 let vn ∈ R2 be the column vector that contains the n-th and (n − 1)-th Fibonacci
number: vn = (Fn , Fn−1 )T . (So v1 = (1, 1)T , v2 = (2, 1)T , v3 = (3, 2)T , v4 = (5, 3)T .) Show that
1 1
with the matrix A =
we have that vn+1 = Avn .
1 0
b) Determine eigenvectors and eigenvalues for A.
c) Find a diagonal matrix D and a matrix P such that D = P−1 AP.
d) Determine a formula for An .
e) Use the result of d) and the fact that vn+1 = An v1 to obtain a formula for Fn for general n.
42∗ ) Show that the characteristic
xn − an−1 xn−1 − · · · − a2 x2 − a1 x − a0 .

0
 1

 0


A= 0
 0

 ..
 .
0
43)
polynomial of the matrix A, given below, is χA (x) =
0
0
1
0
0
..
.
0
0
0
1
0
..
.
0
0
0
0
1
..
.
···
···
···
···
···
..
.
···
0
0
0
0
0
0
a0
a1
a2
a3
a4
..
.
0
1 an−1











Let V = R3 with standard basis S and L:V → V defined by


2 −1
4
 1
4 −3 
S [L]S =
0
0
3
T
a) Determine a basis
h
iB for the L-cyclic subspace W generated by the vector v = (0, 1, 0) .
b) Determine L/W
and the characteristic polynomial for L/W .
B
B
c) Extend B to a basis C of V . h
i
d) Compute C [L]C . (Note that L/W
is a submatrix!)
B
B
e) Using C [L]C determine the characteristic polynomial for L.
44) Determine solutions to the following systems of linear differential equations for the
given initial values:
3 −2
0
x(0) = (1, 1)T
χA (x) = (x − 2)(x + 1)
a)
x (t) =
· x(t),
2 −2
4
5
0
b)
x (t) =
x(0) = (0, 1)T
χA (x) = x2 + 49
· x(t),
−13 −4
46 −42
45) Let A =
. Then χA (x) = (x − 4)(x − 25). Find four different matrices
21 −17
M1 , M2 , M3 , M4 ∈ R2×2 such that Mi2 = A.
Problems marked with a ∗ are bonus problems for extra credit.