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1. What is an eigenvector?2. Find the eigenvectors of the following matrices.(a)21[0(b)-1 1-1(c)1 13. What is a fixed point of a function?4. Find the fixed points of the following discrete time difference equations.(a) Nt+1 (1+rN) Nt(b) Xt+1 (1 + X) Xt 212+3 (Y? - 4Y)(c) Y+15. Find the equilibria of the following differential equations.(a) mx"(t) ax' (t)+ kx(t) = 0y(t)1(b) y'(t) ry(t)K(c) N'(t) AN (t)6. Bonus: for each of the differential equations above, how do values of y(t) (or x(t)fixed points "behave"? (i.e. are they moving towards or away from the fixed points?)or N(t))near the

Question

I need help for problem 2b. Thanks

1. What is an eigenvector?
2. Find the eigenvectors of the following matrices.
(a)
2
1
[
0
(b)
-1 1
-1
(c)
1 1
3. What is a fixed point of a function?
4. Find the fixed points of the following discrete time difference equations.
(a) Nt+1 (1+rN) Nt
(b) Xt+1 (1 + X) Xt 2
12+3 (Y? - 4Y)
(c) Y+1
5. Find the equilibria of the following differential equations.
(a) mx"(t) ax' (t)+ kx(t) = 0
y(t)
1
(b) y'(t) ry(t)
K
(c) N'(t) AN (t)
6. Bonus: for each of the differential equations above, how do values of y(t) (or x(t)
fixed points "behave"? (i.e. are they moving towards or away from the fixed points?)
or N(t))near the
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1. What is an eigenvector? 2. Find the eigenvectors of the following matrices. (a) 2 1 [ 0 (b) -1 1 -1 (c) 1 1 3. What is a fixed point of a function? 4. Find the fixed points of the following discrete time difference equations. (a) Nt+1 (1+rN) Nt (b) Xt+1 (1 + X) Xt 2 12+3 (Y? - 4Y) (c) Y+1 5. Find the equilibria of the following differential equations. (a) mx"(t) ax' (t)+ kx(t) = 0 y(t) 1 (b) y'(t) ry(t) K (c) N'(t) AN (t) 6. Bonus: for each of the differential equations above, how do values of y(t) (or x(t) fixed points "behave"? (i.e. are they moving towards or away from the fixed points?) or N(t))near the

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Step 1

Given: -

0
1
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0 1

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Step 2

To calculate: -

Eigen vectors of the given matrix.

Step 3

Calculation...

Finding the characteristic equation that is, |A -AI from the given matrix
So,
1 0
1
0
(A-I)
1
0 1
0
-2
1
-1 1-
0 A
-2.
1
=(-2)(1-)-(-1) = -2+22 +1
-1 1-2
A-A=22-2+1
(1)
Now, putting (1) equals to 0, then
2-2+1 0
1+3i 1-3i
2 2
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Finding the characteristic equation that is, |A -AI from the given matrix So, 1 0 1 0 (A-I) 1 0 1 0 -2 1 -1 1- 0 A -2. 1 =(-2)(1-)-(-1) = -2+22 +1 -1 1-2 A-A=22-2+1 (1) Now, putting (1) equals to 0, then 2-2+1 0 1+3i 1-3i 2 2

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Tagged in

Math

Algebra

Matrices

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