Q28E There exists a nonzero upper tri... [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: Q28E (page 108)

There exists a nonzero upper triangular 2 脳 2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

Short Answer

Expert verified

The statement is false.

Step by step solution

Proving statement by example

The claim is that there exists an upper triangular 2x2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

First, let A= $[\begin{matrix} a & b \\ 0 & c \end{matrix}]$ from here we see that:

$\begin{array}{l} A^{2} = [\begin{matrix} a & b \\ 0 & c \end{matrix}] [\begin{matrix} a & b \\ 0 & c \end{matrix}] \\ A^{2} = [\begin{matrix} a^{2} & a b + b c \\ 0 & c^{2} \end{matrix}] \end{array}$

Conclusion

Now if $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ this means that: $[\begin{matrix} a^{2} & a b + b c \\ 0 & c^{2} \end{matrix}] = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$

$鈬� a = 0$ but c = 0 under these conditions, A is no longer an upper triangular matrix and thus the statement is false.

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

There exists a nonzero upper triangular 2 脳 2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .

Short Answer

Step by step solution

Proving statement by example

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Mechanics Maths

Calculus

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

91影视

There exists a nonzero upper triangular 2 脳 2 matrix A such that A2=[0000].

Short Answer

Step by step solution

Proving statement by example

Conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Mechanics Maths

Calculus

Statistics

Logic and Functions

Study anywhere. Anytime. Across all devices.

There exists a nonzero upper triangular 2 脳 2 matrix A such that $A^{2} = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ .