CS代考计算机代写 Econometrics Exercise 1 (Ch. 2)

Econometrics Exercise 1 (Ch. 2)
Prof. Dr. Dominik Liebl
Review of Matrix Algebra
Exercise 1
Calculate the following sums and products (as far as they are defined):
Exercise 2
􏱫1−2􏱬 A=3·5 7
􏱫3 1􏱬􏱫6 7 −3􏱬 C= 2 0 −2 −1 4
􏱩 􏱪􏱫3 5􏱬􏱫−3􏱬 􏱫6 3 −3􏱬′􏱫1 −8 −4􏱬 G=−42−97−9 H=412752
􏱫−1 0􏱬􏱩 􏱪
E=1212 F=719
Determine the rank of these matrices (you can use R to solve this exercise):
1 0 3 2 􏱫1 −1􏱬 A=50−1−6, B=−10.
4 0 2 −2
Show that
for any two n × n matrices A and B that have full rank (i.e. rk A = rk B = n).
Exercise 4
Let A and B be n × n matrices with full rank. Calculate
a) (AB)′(B−1A−1)′
b) (A(A−1 + B−1)B)(B + A)−1 (assuming (B + A) is invertible)
Exercise 3
(AB)−1 = B−1A−1
1
􏱩􏱪􏱩 􏱪
B=31+−2−3 􏱫3 7􏱬
D=6+ 1 2 􏱫3􏱬􏱩 􏱪

Exercise 5
Consider the matrix
where x1,x2,…,xn ∈ Rm. Show that
Exercise 6
Show that
X=􏱩x1 x2 ··· xn􏱪′,
n
􏱛 x i x ′i = X ′ X . i=1
nn
􏱛(xi −x)(yi −y)=􏱛(xi −x)yi
i=1 This is the “usefull result” from Chapter 2.
i=1
2