Maths matrices

Maths for economists Problem Set II

due 10/15/15

Each question is worth 2 points. Total is 20 points but 24 points are possible. Show your work.

1 Determinans

Find determinants for the following matrices:

a- W =   1 1 71 1 −3

2 1 1

 

b- B =

 

2 2 1 1 1 1 1 1 1 −1 1 1 1 −1 1 1 1 −1 1 1 1 −1 1 1 1

 

2 System of equations

Consider the following system of equations:

 

2x1 + 2x2 + x3 + 2x4 = b1 2x1 + 2x2 + x3 − 2x4 = b2 x1 + 2x2 − x3 + 2x4 = b3 x1 − 2x2 + x3 + 2x4 = b4

a- Solve the system for b =

 

1 4 7

10

  using the general form x = A−1b

b- Find x1 and x4 for b =

 

2 5 8

11

  using Cramer’s rule.

3 Diagonalization

a- Find the eigenvalues for G =   0 1 11 0 1

1 1 0

 .

1

b- Find the diagonalized expression of G .

c- Let n ∈N. Find the general formula for 3G n .

4 Quadratic forms

a- Show that q1(x , y ) = 2x 2 +4x y +3y 2 and q2(x , y, z ) = 10(x 2 + y 2)+z 2 +2(x y +x z +x y ) are positive definite b- Let n ∈ N. What is the minimum value of n for the quadratic form expressed by matrix (B + n I ) to be

positive definite? (hint: its characteristic polynomial P (λ) can be written P (λ) = (x − (2 + n))2Q (λ))

5 Second order conditions and concavity

a- Find and study the turning points of f (x , y, z ) = x 2 + y 2 + z 2 − x y − x z − 6x . b- Find and study the turning points of f (x , y ) = x 2 + 2x y 2 − x . c- Study the concavity of f (x , y ) = x 4 + y 4 + x 2 + y 2.

2