Q26E In Exercises 17 through 26, find... [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 2: Q26E (page 85)

In Exercises 17 through 26, find all matrices that commute with the given matrixA.
$A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{matrix}]$

Short Answer

Expert verified

The matrix $A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{matrix}]$ is commute with all the matrices of the form $[\begin{matrix} a & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & i \end{matrix}]$ .

Step by step solution

Assuming the matrix

Let the given matrix $A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{matrix}]$ .

Let the matrix commute with matrix A of the form $B = [\begin{matrix} a & d & g \\ b & e & h \\ c & f & i \end{matrix}]$ ,Where is constant number.

Commutative matrix

Now since multiplication of matrix A and B commute then $A B = B A$

$A B = [\begin{matrix} 2 a & 2 d & 2 g \\ 3 b & 3 e & 3 h \\ 4 c & 4 f & 4 i \end{matrix}], B A = [\begin{matrix} 2 a & 3 d & 4 g \\ 2 b & 3 e & 4 h \\ 2 c & 3 f & 4 i \end{matrix}]$

Now

$A B = B A A B = [\begin{matrix} 2 a & 2 d & 2 g \\ 3 b & 3 e & 3 h \\ 4 c & 4 f & 4 i \end{matrix}], B A = [\begin{matrix} 2 a & 3 d & 4 g \\ 2 b & 3 e & 4 h \\ 2 c & 3 f & 4 i \end{matrix}]$

On equating the corresponding elements of obtained matrix, we get equations

role="math" localid="1664213141344" $2 a = 2 a 2 d = 3 d 2 g = 4 g 3 b = 2 b 3 e = 3 e 2 h = 4 h 4 c = 2 c 3 f = 4 f 4 i = 4 i$

On solving these equations, we get

$a = a, b = 0, c = 0, d = 0, e = e, f = 0, g = 0, h = 0, i = i$

Hence, Matrix is $A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{matrix}]$ commute with all the matrix of the form $[\begin{matrix} a & 0 & 0 \\ 0 & e & 0 \\ 0 & 0 & i \end{matrix}]$ .

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

In Exercises 17 through 26, find all matrices that commute with the given matrixA.
$A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{matrix}]$

Short Answer

Step by step solution

Assuming the matrix

Commutative matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

91影视

In Exercises 17 through 26, find all matrices that commute with the given matrixA.A=200030004

Short Answer

Step by step solution

Assuming the matrix

Commutative matrix

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Discrete Mathematics

Mechanics Maths

Geometry

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 17 through 26, find all matrices that commute with the given matrixA.
$A = [\begin{matrix} 2 & 0 & 0 \\ 0 & 3 & 0 \\ 0 & 0 & 4 \end{matrix}]$