Question

1 0 4. Consider the matrices A = 0 +- Alw alcaldo and B o -1010 = 01. Answer the following o 0 2 questions.

$(5) Find all the vectors x and y which satisfy the following simultaneous equations. y = lim {A^ + B” k} n >00 \y\=1. Here, y$

Answer 1

The Eigenvalues and Eigenvectors of are

1.

21-*-*

2.

12 = 1 +V2=

3.

13 = 1 +13=

Therefore we can write

10 -13 0 A= 11 [ 0 0 0 0 0 0 To 1 0 0 0 3 1 ortak 1 00

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The Eigenvalues and Eigenvectors of are

1.

11 =1 V1=

2.

12 = 2 +V2=

3.

$\lambda_3 = \frac{1}{2} \rightarrow \mathbf{v}_3 = \begin{bmatrix} -1 \\ 1 \\ 0 \end{bmatrix}$

Therefore we can write

B = [1 0 -1] [100] 1 0 1 0 2 0 0 1 0 ] [oo i

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Thus we have

0 lim OMIA n-00 70 lim 01] [1 07" To - 43 A" = lim -V3 0 1 0 0 "** [ 1 1 ] LOO 3] [i o o TO 01] [100] [o - 3 A” = lim -13 0 0 1" 0 V3 *** | 1 1 o Lo 03"] [1 öö] To 0 11 Too 01 To - y3 11 A” = -3 0 0 1 0 0 3 3 | 1 1 0 0 0 0 1 0 0 7+00 00 lim 700

Therefore,

lim A" = 0 n-00 olan iki celana lim n +00 A" = lim n+00 A" = 0 1 V3

Similarly,

1 n+001 +00 lim ſi 0 -1] [100]" | = lim 1 0 1 0 2 0 0 0 1 *** LO 1 0] LO o i] [-į o 10 -11 1 0 0 0 B" = lim 1 0 1 0 2" 0 0 1 *** 0 1 0 0 0 - 0 [1 0 -1] [100] B" = 1 0 1 0 0 1 ó ó 1 0 1 0 0 0 0 - 0 n +00 lim n+00 NON-

Therefore,

lim n+00 B" = 0 0 0 1 ON-OF 008-IS-INO Ź o lim n+00 B" =

Therefore,

y = y= 0-808 MOOS-8 + 'DUH

Let

X =

Then

1 y1 = 2 * x2 + 3x12 y2 = + 3 x 22+ V3x. 4 y3 = 1 x 21 +0 X 22 + OC X 13

Since we require that

y = 1

41, 42, 43, <1

This requires that

21 = 13 = 0

Then we have

$\\ y_1 = \frac{x_2}{2} \\ y_2 = \frac{3x_2}{4} \\ y_3 = 0$

Now,

vệ - vì vi vi -( ) -

Thus we have

y= + 13 , X= 1