Dr. Thomas Zehrt
Topics in Advanced Mathematics
Exercise 1
Fundamentals, differentiable functions and Maple
Solve the following problems and hand in your solutions at the beginning of the course
next week. The solutions will be marked.
1. The rank r(A) of a matrix A is the maximal number of linearly independent column
vectors in A. Determine the rank of the following matrix for all values of the real
parameter a.


1 −1 2
0
A =  −1 a −1 a 
2
0
1 −1
2. Verify the Spectral Theorem for symmetric matrices for the following matrix
by finding a matrix P:


1 1 0
A =  1 1 0 .
0 0 2
3. Let A be an m × n matrix and b ∈ Rm . Prove that the set
M = {x ∈ Rn | Ax ≤ b and x ≥ 0}
is convex.
4. Let B be an n × n matrix. Show that the matrix A = BT B is positive semidefinite.
5. (a) Compute the gradients of the function f (x, y) = y 2 + xy at the points (2, 1),
(4, 2) and (6, 3).
(b) Let f (x) = f (x1 , . . . , xn ) be homogeneous of degree d. Prove the following
fact: At each point on a given ray through the origin the gradients of f are
proportional.
6. Prove that if φ is twice countinuously differentiable and φ(x, y) = c defines y as a
twice differentiable function of x, then
y 00 = −
φxx + 2φxy · y 0 + φyy · (y 0 )2
.
φy
7. Let f (x1 , x2 , x3 ) = −x21 + 6x1 x2 − 9x22 − 2x23 . Prove that f is concave and determine
all points with vanishing gradient.
8. Suppose y = f (x) is a production function determining the output y as a function
of the vector x of nonnegative factor inputs, with f (0) = 0. Show that:
(a) If f is concave, then fxi xi (x) ≤ 0 for all i.
(b) If f is concave, then f (λx)/λ is decreasing as a function of λ.
(c) If f is homogeneous of degree 1, then f is not strictly concave.