Department of Economics
Econ 2750A
H.W. #3
Prof. K.C. Tran
Due Date:
Friday, February 27, 2009
1. (a) Given the following matrices:
1  1 0 
A  4 2  2,
5 0  1
 0 3 4
B   1 7 5
 0 1 1
Determine whether A and B are positive definite, negative definite or indefinite
(b) Determine the definiteness of the following quadratic forms:
(i)
q  x12  4 x1 x 2  2 x1 x3  x 22  4 x32  6 x 2 x3
(ii)
q  2 x12  3x 22  5 x32  8 x1 x3
2. Given the following matrix:
(a) Find the eigenvalues of A.
(b) Find the eigenvectors of A.
3. Use Cramer’s rule to solve for x1 and x3 in the following linear system:
 x1  3x2  2 x3  24
x1  x3  6
5 x2  x3  8
4. (a) If matrix A is a 7 9 matrix with three linearly independent rows, that is the rank
of A?
(b) If matrix A is a 12 6 , what is the largest possible rank of A?
(c) Use the property of determinant to show the following:
If A and B are squared matrices with the inverse of B exists, then
BAB 1  A