⇐ 19 )

MATH 2022 Week06Worksheet

MATH 2022 Week 6 Worksheet

Qy Putm = [ ?I )

.

Ca) find

me :3 -

- 5.' all - it -

-

ml ! ] --

Cb) By inspection ,from the previous part ,

what are the eigenvalues ? 17-

(c) Find P,P"

and diagonal D

such that M = PDP- l

,

p =

,p

- '=

D=

(d) Find the characteristic polynomial

Xcx ) -- detent - ml =/ IT

=

Ce ) What are the roots of XCX ) ?

X =

Cf I find a general formula for M"

:

M"

= ( PDP - 1)n

= p D~

p- I

=

⑨ Thus find MY =

QY Putm= f I!! ) .

⑨ find

MEI -. fit

mail --

Mf !) -

-

Cb) By inspection ,what are the eigenvalues?

IT-

(c) Find P and diagonal D such

that M = PDP- l

:

P = D=I

(d) Find p- I

:

Ce ) find a general formula for M"

:

M"

=

Cfl Thus find MY =

0-31Put

m = [ II ].

Ca ) Find surd expressions for the

eigenvalues :

Cb) What is the dominant eigenvalueto 3 decimal places ?

x,

= IT-

e) What is the smaller eigenvalueto 3 decimal places ?

t.

-- IT

-

Cd) findM

- I=

(e) Put I .= [ ! ] ,

Ike ,= M In ( kzo )

1st entry of Inrn =

-

1st entry of Ih ,

2nd entry of InSh = -

1st entry of In

Complete the following table to 3 d. p .

To Ii In Is Iy It Is It

lol its

Te I 7-

Sk 3 2-143

Cf) find M [ I , ) =

(g) Read off the dominant eigenvalue I M-

(he) Put To = [ ! ] ,

Int,

= M- '

In = ( III ) Inf

1st entry of Ikk

=-

1st entry of In - I

=

2nd entry of InUk -

1st entry of In

Complete the following table to 3 d. p .

So I , In Is IT

lol Csi:Dth - 2

Uk - o . 75

( i ) find M [

lay) =

( j ) Read off the negative eigenvalue, 17of M

-

Q4/Suppose that M is an invertible

matrix with eigenvector I correspondingto eigenvalue X

.

(a) Prove that X to .

Proof :

Cb ) Prove that I is an eigenvectorfor M

"

corresponding to eigenvalue 5'

.

Proof :