Exercise 8.5.5 Let A =-206-3 1 6-3 0 7Find a square root of A.

Take the determinant equal to zero.Write all three eigenvalues in the diagonal matrix D

Answered: Exercise 8.5.5 Let A - -206 -3 16 -3 0…

Algebra and Trigonometry (6th Edition)

6th Edition

ISBN:9780134463216

Author:Robert F. Blitzer

Publisher:Robert F. Blitzer

ChapterP: Prerequisites: Fundamental Concepts Of Algebra

Section: Chapter Questions

Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...

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Question

Exercise 8.5.5 Let A =
-206
-3 1 6
-3 0 7
Find a square root of A.

Expert Solution

Step 1: Diagonal matrix

$A equals open parentheses table row cell negative 2 end cell 0 6 row cell negative 3 end cell 1 6 row cell negative 3 end cell 0 7 end table close parentheses left parenthesis A minus x I right parenthesis thin space thin space i s comma open parentheses table row cell negative 2 end cell 0 6 row cell negative 3 end cell 1 6 row cell negative 3 end cell 0 7 end table close parentheses minus x open parentheses table row 1 0 0 row 0 1 0 row 0 0 1 end table close parentheses open parentheses table row cell negative 2 minus x end cell 0 6 row cell negative 3 end cell cell 1 minus x end cell 6 row cell negative 3 end cell 0 cell 7 minus x end cell end table close parentheses$

Take the determinant equal to zero.

$open parentheses negative 2 minus x close parentheses open parentheses negative x plus 1 close parentheses open parentheses negative x plus 7 close parentheses minus 0 times open parentheses 3 x minus 3 close parentheses plus 6 times thin space 3 open parentheses negative x plus 1 close parentheses equals 0 open parentheses negative 2 minus x close parentheses open parentheses negative x plus 1 close parentheses open parentheses negative x plus 7 close parentheses plus 6 times 3 open parentheses negative x plus 1 close parentheses equals 0 open parentheses negative 2 minus x close parentheses open parentheses negative x plus 1 close parentheses open parentheses negative x plus 7 close parentheses plus 18 open parentheses negative x plus 1 close parentheses equals 0 open parentheses negative x plus 1 close parentheses open parentheses open parentheses negative 2 minus x close parentheses open parentheses negative x plus 7 close parentheses plus 18 close parentheses equals 0 open parentheses negative x plus 1 close parentheses open parentheses x squared minus 5 x minus 14 plus 18 close parentheses equals 0 open parentheses negative x plus 1 close parentheses open parentheses x squared minus 5 x plus 4 close parentheses equals 0 open parentheses negative x plus 1 close parentheses open parentheses x minus 1 close parentheses open parentheses x minus 4 close parentheses equals 0 open parentheses x minus 1 close parentheses open parentheses x minus 1 close parentheses open parentheses x minus 4 close parentheses equals 0 open parentheses x minus 1 close parentheses squared open parentheses x minus 4 close parentheses equals 0 x equals 1 comma thin space x equals 4 x equals 1 thin space with thin space multiplicity thin space of thin space 2 comma thin space x equals 4$

Write all three eigenvalues in the diagonal matrix D

$D equals open parentheses table row 1 0 0 row 0 1 0 row 0 0 4 end table close parentheses$