Exercise 2.14. Let A € Mmn (R). For r = {1,...,m} and s € {1,...,m}, let Irs €Mm (R) be the matrix whose rs-component is 1 and all the other ones are equal to 0.Answer the following questions with words :1. What is Is A ?2. For rs, what is (Irs + Isr) A ?3. For rs, what is (1m + Irs + Isr - Irr - Iss) A ?4. For rs, what is (1m+cIrs) A, for some c ER ?

Given that .1) 2) for Now Hence

Answered: Exercise 2.14. Let A € Mmn (R). For r =…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Exercise 2.14. Let A € Mmn (R). For r = {1,...,m} and s € {1,...,m}, let Irs €
Mm (R) be the matrix whose rs-component is 1 and all the other ones are equal to 0.
Answer the following questions with words :
1. What is Is A ?
2. For rs, what is (Irs + Isr) A ?
3. For rs, what is (1m + Irs + Isr - Irr - Iss) A ?
4. For rs, what is (1m+cIrs) A, for some c ER ?

Expert Solution

Step 1: Elementary matrix Operations

Given that $A space element of M subscript m n end subscript open parentheses straight real numbers close parentheses a n d space I subscript r s end subscript element of M subscript m open parentheses straight real numbers close parentheses space d e n o t e space a space m cross times m space r e a l space m a t r i x space w h o s e space r s space c o m p o n e n t space i s 1 space a n d space o t h e r space c o m p o n e n t space i s space 0$ .

1) $open parentheses I subscript r s end subscript A close parentheses subscript i j end subscript equals open curly brackets table row cell a subscript s j end subscript space comma end cell cell i equals s comma for all j end cell row cell 0 space comma end cell cell i not equal to s comma for all j end cell end table close$

2) for $r not equal to s comma left parenthesis I subscript r s end subscript plus I subscript s r end subscript right parenthesis A equals I subscript r s end subscript A plus I subscript s r end subscript A$

Now $open parentheses I subscript r s end subscript A close parentheses subscript i j end subscript equals open curly brackets table row cell a subscript s j end subscript end cell cell for all j end cell row 0 cell i not equal to s comma for all j end cell end table close space a n d space open parentheses I subscript s r end subscript A close parentheses subscript i j end subscript equals open curly brackets table row cell a subscript r j end subscript end cell cell i equals r comma for all j end cell row 0 cell i not equal to r comma for all j end cell end table close$

Hence $open parentheses open parentheses I subscript r s end subscript plus I subscript s r end subscript close parentheses A close parentheses subscript i j end subscript equals open parentheses I subscript r s end subscript A close parentheses subscript i j end subscript plus open parentheses I subscript s r end subscript A close parentheses subscript i j end subscript equals open curly brackets table row cell a subscript r j end subscript comma w h e n space i equals r space a n d space a subscript s j end subscript space w h e n space i equals s end cell row cell 0 comma o t h e r w i s e end cell end table close$