Exercise: Let A = 35 2 3 3 1 2 32 1 2 1 0 3 -9 18 -57 18 -2 14 -28 -1 2 -4 6 12 -2 -30 -10 5 -9 -2 2 -4 18 14 6 12 2 -57 -28 1. -2 -30 18 -1 -10 5 The matrix A satisfies -1 2 0 1 -1 2 2 5 1 TOT 0 -1 and 0 -3 3 11 1000 02 0 -4 0 0 0 0 0 where the matrix multiplying A on the left is the inverse of the one multiplying A on the right. The eigenvalues of A are 0 0 9]
Exercise: Let A = 35 2 3 3 1 2 32 1 2 1 0 3 -9 18 -57 18 -2 14 -28 -1 2 -4 6 12 -2 -30 -10 5 -9 -2 2 -4 18 14 6 12 2 -57 -28 1. -2 -30 18 -1 -10 5 The matrix A satisfies -1 2 0 1 -1 2 2 5 1 TOT 0 -1 and 0 -3 3 11 1000 02 0 -4 0 0 0 0 0 where the matrix multiplying A on the left is the inverse of the one multiplying A on the right. The eigenvalues of A are 0 0 9]
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.3: Spanning Sets And Linear Independence
Problem 42EQ
Related questions
Question
![Recall: An n x n matrix A is diagonalizable if there exists an invertible n x n matrix
P and a diagonal n x n matrix D and such that
P-¹ AP = D
Keep in mind this equation when answering the following:
Exercise: Let A =
لیا
3
5
2 3
3 3
1 2
3
2 1 2
2 1 0 1
-9
18
-57
18
-9
18
-2 2
-4
14 6
12
-28
-2
-30
-1 -10 5
-2 2 -4
14
6 12
-2 -30
-1 -10 5
-57-28
The matrix A satisfies
[-3 -3 -6]
-1 2 0 -1
0 1 -1
2 1
-1
2 -5 1
0
-3
3
and
=
18
where the matrix multiplying A on the left is the inverse of the one multiplying A on
the right.
The eigenvalues of A are
1 0
02
00
0 0
0 0
0 0
-4 0
0 9]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aa27a28-efdb-4d0f-aad3-266fe3536dfa%2F1783f05d-b53a-43df-a236-1f979e49f263%2F7o2tvop_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Recall: An n x n matrix A is diagonalizable if there exists an invertible n x n matrix
P and a diagonal n x n matrix D and such that
P-¹ AP = D
Keep in mind this equation when answering the following:
Exercise: Let A =
لیا
3
5
2 3
3 3
1 2
3
2 1 2
2 1 0 1
-9
18
-57
18
-9
18
-2 2
-4
14 6
12
-28
-2
-30
-1 -10 5
-2 2 -4
14
6 12
-2 -30
-1 -10 5
-57-28
The matrix A satisfies
[-3 -3 -6]
-1 2 0 -1
0 1 -1
2 1
-1
2 -5 1
0
-3
3
and
=
18
where the matrix multiplying A on the left is the inverse of the one multiplying A on
the right.
The eigenvalues of A are
1 0
02
00
0 0
0 0
0 0
-4 0
0 9]
![-3 -3 -6
Exercise: Let B = 2
47
4 2
27
The matrix B satisfies
-1
-6
-1 -1
-3 -3
1 -3 1
-2 -1 -2 2
4
2
-2 1
0
-4 -2 -5 4 2 7 02 -1
A basis for the eigenspace E₂ is BE2
where the matrix multiplying B on the left is the inverse of the one multiplying B on
the right.
The eigenvalues of B are λ = 2 and λ = 3.
3 0 0]
020
0 0 3.
A basis for the eigenspace E3 is BE3 =
HI
(88)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0aa27a28-efdb-4d0f-aad3-266fe3536dfa%2F1783f05d-b53a-43df-a236-1f979e49f263%2F4dihhx6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:-3 -3 -6
Exercise: Let B = 2
47
4 2
27
The matrix B satisfies
-1
-6
-1 -1
-3 -3
1 -3 1
-2 -1 -2 2
4
2
-2 1
0
-4 -2 -5 4 2 7 02 -1
A basis for the eigenspace E₂ is BE2
where the matrix multiplying B on the left is the inverse of the one multiplying B on
the right.
The eigenvalues of B are λ = 2 and λ = 3.
3 0 0]
020
0 0 3.
A basis for the eigenspace E3 is BE3 =
HI
(88)
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