Question

For what values of x if any will the matrix A=[3 0 0; 0 x 2; 0 2 x] have at least one repeated eigenvalue?

Answer 1

does it have something to do with symmetry?

Answer 2

intuition tells me 2 or -2....but im working it out to see if thats the case

Answer 3

| 3-c 0 0|
|0 x-c 2|
|0 2 x-c|
=(3-c)[(x-c)^2-4]
3-c=0=>c=3
(x-c)^2=4 => x-c=2 and x-c=-2=> x=2+c and x=-2+c
....

Answer 4

x=2+3=5 or
x=-2+3=1

Answer 5

Why do you have c's in your work?

Answer 6

those are his eigenvalues (lambda)

Answer 7

c is the eigenvalue

Answer 8

her*

Answer 9

sry sry, those are her eigenvalues (lambda)

Answer 10

someone might want to check my x's
is that right?

Answer 11

Oh haha okay but I got (c-3)(c^2-2cx+x^2-4)

Answer 12

im on it, one sec

Answer 13

i think it is right

Answer 14

its it :)

Answer 15

just finished checking it, i wish i knew a fast way to do problems like these >.>

Answer 16

one question though once you get the characteristic polynomial how'd you know what values of x works?

Answer 17

first i found c

Answer 18

or lambda whatever you want to name your eingenvalues

Answer 19

lol

Answer 20

which would have a x value in it right

Answer 21

then i used c to find the x's
the other other factor of the characteristic polynomial was (x-c)^2-4
i set this=0
x-c=2
x-c=-2
but c=3
so
one x is 5
another possible x is x=1

Answer 22

det(A-cI)=0

Answer 23

Oh okay make sense now thanks for all the help

Answer 24

nice problem :)

Answer 25

very interesting, hadnt seen one like it before.