Question

(a) Show that the characteristic polynomial χA(λ) of the matrix

(0 b 0)
(b a b)
(0 b a)

is given by χA(λ) = −λ3 + 2aλ2 + (2b2 − a2)λ − b2a.

Please show work...kind regards

Answer 1

subtract L from the diagonal and take the determinant ... are the steps if i recall them correctly

Answer 2

okay....amistre . I will try what you said....If I get stuck I'll shout for help :)

Answer 3

i hope you know the shortcut for the determiant of a 3x3 :)

Answer 4

I use row reduction

Answer 5

row reduce and multiply the diagonal? i remember that being a suitable method

Answer 6

yes...I am not sure if that works here...hmmm...I have done the first part ...what would be best method to calculate determinant

Answer 7

"best" method .... is highly subjective. The sure method to me would be the standard run down the most zeroed col/row applying submatrixes

Answer 8

yes, i will have to revise that as my fav method is row reduction

Answer 9

(-L b 0 )
( b a-L b )
( 0 b a-L)

\[-L \begin{vmatrix}a-L&b\\b&a-L\end{vmatrix}-b\begin{vmatrix}b&0\\b&a-L\end{vmatrix}+0|...|\]

Answer 10

\[-L\ [(a-L)^2-b^2] - b\ [b(a-L)-0]\]

Answer 11

and simplify to your hearts content :)

Answer 12

Two questions : where does the b come from and what is the 0 at the end (is it needed) . Thank you for your help. You're amazing.

Answer 13

using the standard method of getting a determinant

i ran down the first column in a +-+- fashion to get the scalars

+(-L) |...| - (b) |....| + (0) |....|

Answer 14

yes

Answer 15

this is much easier than row reduction

Answer 16

cross out the row/col of the scalar you are using to determine the submatrix to apply the scalar to

Answer 17

yes