Linear Optimization (1:640:354:05)
Spring 1999, Professor Ben-Israel
Assignment 2: Matrix Algebra
Date: Tuesday, February 2
Due: Tuesday, February 9
This assignment has five problems, of equal weight.
1 Consider the three elementary
row operations on a matrix with n rows,
- E1(n,i,j) swap rows i and j
- E2(n,i,a) multiply row i by a
- E3(n,i,j,a) add a*row i to row j
and their matrix representations.
(a) Show that these operations are not independent, by expressing
E1(n,i,j) in terms of the other two operations.
(b) Verify your answer by using the matrix representations of the three
operations, and explain why det(A1)= -1.
2 Consider the matrix
A with
A[i,j] := 4*(i-1) + j , i,j = 1,..,4
(a) Find the general solution of Ax=b, or determine that the system is
inconsistent, for the RHS's
b := [1,1,1,1] and b := [2,-3,1,4].
(b) Find a basis for the null space of A.
(c) Find a basis for the range of A, and determine a condition for the
consistency of Ax=b in terms of the RHS b.
3 Text, p. 91, Exercise 8.
Hint: The problem can be formulated without the
variable x3.
4 Text, p. 99, Exercise 2.
5 Text, p. 99, Exercise 4.
Return to the HW Assignments page