Q100E Find a nonzero聽n脳n matrix A聽w... [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: Q100E (page 103)

Find a nonzero $n 脳 n$ matrix Awith identical entries such that $A^{2} = A$ .

Short Answer

Expert verified

A nonzero $n 脳 n$ matrix A with identical entries such that $A^{2} = A$ is

$A = {(\begin{matrix} \frac{1}{n} & 鈥� & \frac{1}{n} \\ 鈰� & 鈰� & 鈰� \\ \frac{1}{n} & 鈥� & \frac{1}{n} \end{matrix})}_{n 脳 n}$

Step by step solution

Definition of Idempotent matrix

A square matrix A of order $n 脳 n$ is said to be idempotent if (and only if) $A^{2} = A$ .

Considering an example

Let us take a matrix A of order $2 脳 2$ such that $A = {[\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{matrix}]}_{2 脳 2}$ .

Find the matrix $A^{2}$ :

$A^{2} = [\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{matrix}] [\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{matrix}] = [\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{matrix}] 鈬� A^{2} = A$

Finding ann×n matrix such that A2=A

Since we have seen, from step 2, that for a 2 x 2 matrix A we get $A^{2} = A$ such that

$A = {[\begin{matrix} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \end{matrix}]}_{2 脳 2}$

$鈬�$ This is true for all $n 脳 n$ matrix A such that $A^{2} = A$ where $A = {(\begin{matrix} \frac{1}{n} & 鈥� & \frac{1}{n} \\ 鈰� & 鈰� & 鈰� \\ \frac{1}{n} & 鈥� & \frac{1}{n} \end{matrix})}_{n 脳 n}$

Final Answer

A nonzero n 脳 n matrix A with identical entries such that $A^{2} = A$ is:

$A = {(\begin{matrix} \frac{1}{n} & 鈥� & \frac{1}{n} \\ 鈰� & 鈰� & 鈰� \\ \frac{1}{n} & 鈥� & \frac{1}{n} \end{matrix})}_{n 脳 n}$ .

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

Find a nonzero $n 脳 n$ matrix Awith identical entries such that $A^{2} = A$ .

Short Answer

Step by step solution

Definition of Idempotent matrix

Considering an example

Finding ann×n matrix such that A2=A

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Mechanics Maths

Calculus

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

91影视

Find a nonzeron脳n matrix Awith identical entries such thatA2=A.

Short Answer

Step by step solution

Definition of Idempotent matrix

Considering an example

Finding ann×n matrix such that A2=A

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Logic and Functions

Mechanics Maths

Calculus

Probability and Statistics

Pure Maths

Study anywhere. Anytime. Across all devices.

Find a nonzero $n 脳 n$ matrix Awith identical entries such that $A^{2} = A$ .