Riva – Linear Algebra
Chapter 4 Review
1 3 5
1) Let 𝐴 = [3 5 7 ].
7 9 11
a. Compute det A by cofactor expansion along any row or column
b. Compute det A by first reducing A to upper triangular form
1
5 −2
2) Let 𝐴 = [
]
−4 4
a. Find the characteristic polynomial of A
b. Find all of the real eigenvalues of A
c. Find a basis for each eigenspace.
1 1 −1
3) Let 𝐴 = [ 0 2 0 ]
−1 1 1
a. Find the characteristic polynomial of A
b. Find all of the real eigenvalues of A
c. Find a basis for each eigenspace.
d. Find the algebraic and geometric multiplicity of each eigenvalue.
2 1 1 1
4) Let 𝐴 = [0 1 2 3]
0 0 3 3
0 0 0 2
a. Find the characteristic polynomial of A
b. Find all of the real eigenvalues of A
c. Find a basis for each eigenspace.
d. Find the algebraic and geometric multiplicity of each eigenvalue.
1 −1 2
5) Find all values of k for which |1 1
𝑘 | = 0.
2 4 𝑘2
3
13 −60 −45
6) Show that [−1] is an eigenvector of [−5 18
15 ] and find the
2
10 −40 −32
corresponding eigenvalue
7) Show that 𝜆 = −2 is an eigenvalue of 𝐴 = [
corresponding to this eigenvalue.
2
8) Find all of the eigenvalues of 𝐴 = [
−1
corresponding eigenspaces.
2 2
] and find one eigenvector
2 −1
1
]. Give bases for each of the
2
Riva – Linear Algebra
𝑘
9) Find all values of k for which 𝐴 = [𝑘 2
0
𝑘
4
𝑘
0
𝑘 2 ] is invertible.
𝑘
10)Find the inverse coefficient matrix and use it to solve the given linear system
𝑥+𝑦−𝑧 = 1
𝑥+𝑦+𝑧 = 2
𝑥−𝑦 =3
11)Find the eigenvalues and corresponding eigenspaces of the upper triangular
𝑎 𝑏
matrix 𝐴 = [
].
0 𝑑
−5 −6 3
12)Let 𝐴 = [ 3
4 −3].
0
0 −2
a. Find the characteristic polynomial of A
b. Find all of the eigenvalues of A
c. Find a basis for each of the eigenspaces of A
d. Determine whether A is diagonalizable. If A is not diagonalizable,
explain why. If A is diagonalizable, find an invertible matrix P and a
diagonalizable matrix D such that 𝑃−1 𝐴𝑃 = 𝐷.
1
13)Let A be a 2x2 matrix with eigenvalues 𝜆1 = 2 and 𝜆2 = −1 and
3
1
1
corresponding eigenvectors 𝑣⃗1 = [ ] and 𝑣⃗2 = [ ]. Find 𝐴−5 [ ].
1
−1
7
1
1
1
14)Let A be a 3x3 matrix with eigenvectors 𝑣⃗1 = [0] , 𝑣⃗2 = [1], and 𝑣⃗3 = [1]
0
0
1
1
1
corresponding to eigenvalues 𝜆1 = − 3 , 𝜆2 = 3, and 𝜆3 = 1, respectively, and
2
𝑥⃗ = [1]. Find 𝐴20 𝑥⃗.
2
1 2 1
15)Determine whether 𝐴 = [0 1 0] is diagonalizable. If so, find an invertible
2 0 2
matrix P and a diagonal matrix D such that 𝐴 = 𝑃𝐷𝑃−1 .
1 1
16)Find all (real) values of k for which 𝐴 = [
] is diagonalizable.
0 𝑘
Riva – Linear Algebra
17)Suppose a car rental agency has three locations in Ottawa: a downtown
location, an east end location, and a west end location. The agency has a
group of delivery drivers that serve all three locations. The agency’s
statistician has determined the following:
- Of the calls to the Downtown location, 30% are delivered in
Downtown area, 30% are delivered in the East end, and 40% are
delivered in the West end.
- Of the calls to the East end location, 40% are delivered in
Downtown area, 40% are delivered in the East end, and 20% are
delivered in the West end.
- Of the calls to the West end location, 50% are delivered in
Downtown area, 30% are delivered in the East end, and 20% are
delivered in the West end.
After making a delivery, a driver goes to the nearest location to make the
next delivery. This way, the location of a specific driver is determined only by
his or her previous location. We assume that each delivery person takes the
same amount of time to make a delivery and get to their next location. The
company begins each day with 12 drivers at each location.
a. Find the transition matrix and set up the equation for determining the
number of drivers at each location after 1 delivery.
b. How many drivers will be at each location after 2 deliveries.
c. How many drivers will be at each location after 10 deliveries.
d. What is the steady state matrix?
18)The weather in Seattle is a major concern of the population. An unofficial
study of the weather in the city in early spring yielded the following
observations:
- It is almost impossible to have two nice days in a row
- If today is a nice one day, it just as likely to have snow or rain the
next day
- If it snows or rains on one day, there is a 50% chance that the
weather will be the same the next day.
- If there is a change from snow or rain, only half of the time is this a
change to a nice day.
a. Write the transition matrix to model this system
b. If it is nice today, what is the probability of being nice after one week?
c. Find the long term behavior of the weather. Explain in words what
your answer tells you.