Q37E If B is a diagonal 3脳3聽matrix,... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q37E (page 177)

If B is a diagonal $3 脳 3$ matrix, what are the possible dimensions of the space V of all $3 脳 3$ matrices A that commute with B?Use Exercises 35 and 36 as a guide.

Short Answer

Expert verified

The possible dimensions of the space V of all $3 脳 3$ matrices A that commute with B are $位 = 3, 5 or 9$

Step by step solution

Explanation for the dimension of the space

If a linear space has a basis with n elements then all other bases of V, consists of n elements as well.

Also we say that n is the dimension of V

$\dim (V) = n$

Solution for the dimension of the basis

Consider, B is a diagonal with $3 脳 3$ matrices.

Assume $B = [\begin{matrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 9 \end{matrix}]$ is the $3 脳 3$ diagonal matrix of B

$|位滨 - B| = 位 [\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}] - [\begin{matrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 9 \end{matrix}] |位滨 - B| = 0 [\begin{matrix} 位 & 0 & 0 \\ 0 & 位 & 0 \\ 0 & 0 & 位 \end{matrix}] - [\begin{matrix} 3 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 9 \end{matrix}] = 0 [\begin{matrix} 位 - 3 & 0 & 0 \\ 0 & 位 - 5 & 0 \\ 0 & 0 & 位 - 9 \end{matrix}] = 0$

Further simplify as follows

$(位 - 3) (位 - 5) (位 - 9) = 0 位 = 3, 5 or 9$

Thus, the possible dimensions of the space V of all $3 脳 3$ matrices A that commute with B are $位 = 3, 5 or 9$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

If B is a diagonal $3 脳 3$ matrix, what are the possible dimensions of the space V of all $3 脳 3$ matrices A that commute with B?Use Exercises 35 and 36 as a guide.

Short Answer

Step by step solution

Explanation for the dimension of the space

Solution for the dimension of the basis

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Discrete Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.

91影视

If B is a diagonal 3脳3matrix, what are the possible dimensions of the space V of all 3脳3matrices A that commute with B?Use Exercises 35 and 36 as a guide.

Short Answer

Step by step solution

Explanation for the dimension of the space

Solution for the dimension of the basis

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Logic and Functions

Discrete Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Geometry

Study anywhere. Anytime. Across all devices.

If B is a diagonal $3 脳 3$ matrix, what are the possible dimensions of the space V of all $3 脳 3$ matrices A that commute with B?Use Exercises 35 and 36 as a guide.