Elementary Matrices In Exercises 1-8, determine whether the matrix is elementary. If it is, state the elementary row operation used to produce it. Finding the Inverse of an Elementary Matrix In Exercises 19-24, find the inverse of the elementary matrix. 1. 2.! 19. [ l 20. To 3. 4. 21. 0 22. 0 10 -3 2 5. 0 6. 0 2 23. 0 24. 7. 00 1000 - --
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.
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Chapter 2 Matrices
2.4 Exercises Se Cuichat com tor woted ou solutona to odd numbered enercises.
Elementary Matrices In Exercises 1-8, determine
whether the matrix is elementary. If it is, state the
elementary row operation used to produce it.
Finding the Inverse of an Elementary Matrix
In Exercises 19-24, find the inverse of the elementary
matrix.
1.
2.
lo
20. 5
19.
3.
12
[o
21. 0
4.
ol
[2
5. 0
22. 0
0 -3
0]
6. 0
[2
[A
23. 0 1
0 0
24.
.
0 0
7.
-5
1
0 0 0
0 0
Finding the Inverse of a Matrix In Exercises 25-28,
find the inverse of the matrix using elementary matrices.
1.
8.
3 -2
25.
26.
0 -3
0 -1
27. 0 6
6 -1
Finding an Elementary Matrix In Exercises 9-12,
let A, B, and C be
28. 0 2
4
2
-1
2
B =
0.
Finding a Sequence of Elementary Matrices
In Exercises 29-36, find a sequence of elementary
matrices whose product is the given nonsingular matrix.
2
21. and
-1
20. l
31. [;
30. ol
4
29.
C =
-1
2
32.
9. Find an elementary matrix E such that FA - R
10. Find an elementary matrix E such that FA - C
I1. Find an elementary matrix F such that FR- A
12. Find an elementary matrix F such that FC- A
[-2
31
2
33. -1 3
34. 2
5
6.
3
4
[4
2
Finding a Sequence of Elementary Matrices
In Exercises 13-18, find a sequence of elementary matrices
that can be used to write the matrix in row-echelon form.
0 -1
3
35.
0 0
36. o
0 -1
0 0 -2
2
2
| -1
3 -3
13. 0 1 ?1
10 -
14. I -1
2 -2
37. Writing Is the product of two elementary matrices
always elementary? Explain.
38. Writing E is the elementary matrix obtained by
interchanging two rows in A is an x matrix.
(a) How will FA compare with A? (b) Find E.
2
2
-2 -1
0]
[1
8 -4 16. 2
3
3
15.
4
5 -1
-4
2
-6
12
8
-2
-2
3 -4
I -2
-6
39. Use elementary matrices to find the inverse of
3
17.
-3
3
18.
2
4
[1
A-0
2
5 -1
- 1
2 -2
8 -5
ongm
2.4 Exercises
83
40. Use elementary matrices to find the inverse of
53. Determine a and h such that A is idempotent.
A-0
1||o
54. Guided Proof Prove that A is idempotent if and only
if A is idempotent.
Getting Started: The phrase "if and only if" means that
you have to prove two statements:
1. If A is idempotent, then A' is idempotent.
2. If A' is idempotent, then A is idempotent.
(i) Begin your proof of the first statement by assuming
that A is idempotent.
(ii) This means that A- A.
(iii) Use the properties of the transpose to show that A
is idempotent.
(iv) Begin your proof of the second statement by
assuming that A" is idempotent.
True or False? In Exercises 41 and 42, determine
whether each statement is true or false. If a statement
is true, give a reason or cite an appropriate statement
from the text. Ifa statement is false, provide an example
that shows the statement is not true in all cases or cite an
appropriate statement from the text.
41. (a) The identity matrix is an elementary matrix.
(b) If E is an elementary matrix, then 2E is an
elementary matrix.
(c) The inverse of an elementary matrix is an
elementary matrix.
42. (a) The zero matrix is an elementary matrix.
(b) A square matrix is nonsingular when it can be
written as the product.of elementary matrices
55. Proof Prove that ifA is an nXH matrix that is
idempotent and invertible, then A -](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc074fc7e-4e48-46f1-bfc4-aee6ba4e6262%2Feea0f418-653b-417c-b6c6-b72939d8445b%2Fte5u17c_processed.jpeg&w=3840&q=75)
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