Q26E State true or false, there exist... [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: Q26E (page 200)

State true or false, there exist a basis of ${鈩�}^{2 脳 2}$ that consist of four invertible matrices.

Short Answer

Expert verified

Therefore, the statement is True.

Step by step solution

Determine the set.

Consider a $2 脳 2$ matrix $A$ from the space $V$ that commute with all the element of $S$ where $S = {B 鈭� M^{2 脳 2} AB = BA}$ .

Determine the basis.

Consider a set $\{[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]\}, \{[\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}]\}, \{[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]\}, \{[\begin{matrix} 0 & 1 \\ 2 & 0 \end{matrix}]\},$ .

The determinant of $[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$ is defined as follows.

localid="1659436287635" $d e t \{[\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}]\} = 1$

The determinant of $[\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}]$ is defined as follows.

$d e t \{[\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}]\} = 2$

The determinant of $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$ is defined as follows.

$d e t \{[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]\} = - 1$

The determinant of $[\begin{matrix} 0 & 1 \\ 2 & 0 \end{matrix}]$ is defined as follows.

$d e t \{[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]\} = - 2$

Therefore, the set $\{[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]\}, \{[\begin{matrix} 1 & 0 \\ 0 & 2 \end{matrix}]\}, \{[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]\}, \{[\begin{matrix} 0 & 1 \\ 2 & 0 \end{matrix}]\},$ is linear independence and the each matrix is invertible.

Hence, there exist a basis of that consist of four invertible matrices.

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

State true or false, there exist a basis of ${鈩�}^{2 脳 2}$ that consist of four invertible matrices.

Short Answer

Step by step solution

Determine the set.

Determine the basis.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Logic and Functions

Decision Maths

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

91影视

State true or false, there exist a basis of 鈩�2脳2that consist of four invertible matrices.

Short Answer

Step by step solution

Determine the set.

Determine the basis.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Calculus

Logic and Functions

Decision Maths

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

State true or false, there exist a basis of ${鈩�}^{2 脳 2}$ that consist of four invertible matrices.