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 16 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