prior to Exercises 15–18 in Section 3.1. [Note: Finding the characteristic polynomial of a 3 x 3 matrix is not easy to do with just row operations, because the variable A is involved.] 0 1 4 -1 4 3 1 0 5 1 9. 10. 1 2 -2 7
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
![49
280 CHAPTER 5 Eigenvalues and Eigenvectors
prior to Exercises 15-18 in Section 3.1. [Note: Finding the
characteristic polynomial of a 3 x 3 matrix is not easy to do with
just row operations, because the variable 2 is involved.]
22. a. If A is 3 x 3
the volume c
b. det AT = (–
4
0 -1
1
1
c. The multipli
of A is calle
9.
4 -1
10.
5
1
-2
value of A.
3
0.
d. A row repla
eigenvalues
-1
2
11.
2
1
4
12.
3.
1
1
4
0.
1
A widely used met
matrix A is the QR e
gorithm produces a s
come almost upper E
the eigenvalues of
matrix similar to A
6 -2
4
0 17
13.
-2
9.
14.
-1
0.
4
8
3
For the matrices in Exercises 15-17, list the real eigenvalues,
repeated according to their multiplicities.
and R1 is upper tria
A1 = R1Q1, which
A2 = R2Q2, and so
the more general res
2
3.
6.
0.
15.
16.
-2
5
-5
23. Show that if A
3
0.
A1 = RQ.
-5
1
24. Show that if A
17.
8.
0.
0 -7
1
.6
25. Let A =
.4
A is the stocl
1
9 -2
3
18. It can be shown that the algebraic multiplicity of an eigen-
value 1 is always greater than or equal to the dimension of the
eigenspace corresponding to 2. Find h in the matrix A below
such that the eigenspace for 1 = 4 is two-dimensional:
tion 4.9.]
a. Find a basi
tor v2 of A
b. Verify that
3
h
c. For k = 1,
A =
and write a
0.
4
14
increases.
19. Let A be an n x n matrix, and suppose A has n real eigenval-
ues, 21,..., An, repeated according to multiplicities, so that
26. Let A =
0063
9233
3330
5200
4000
5304](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fee22cf2f-b974-4b00-a3cf-09b388e7d65d%2Fad026b84-0df6-41e6-b8e3-e5da6ef546f4%2Fujf6qmb_processed.jpeg&w=3840&q=75)
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