640:354:01, 01/29/07
Quiz #1, with solutions
1.
What is the rank of the following matrix? (Justify!)
A
=
0
1
2
3
1
2
3
4
1
1
1
1
Solution:
The reduced row echelon form of
A
is
A
=
1
0

1

2
0
1
2
3
0
0
0
0
.
The rank of the matrix is equal to the number of nonzero rows in its reduced row echelon
form. Therefore, the rank of
A
is 2.
2.
Write an equivalent LP in standard form:
Minimize
z
= 2
x
1
+ 3
x
2

x
3
, subject to
x
1

10
x
2
+
x
3
≥
4
x
1
+
x
3
=
2
x
1
≥
0
, x
2
≥
0
Solution:
LP in standard form is
“Maximize
c
T
x
subject to
Ax
≤
b, x
≥
0
.”
There are three things to
be done:
1. Convert minimization into maximization:
Minimize
z
= 2
x
1
+ 3
x
2

x
3
becomes
Maximize

2
x
1

3
x
2
+
x
3
2. Make all inequalities and equations (except nonnegativity constraints) into inequalities
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 Spring '08
 VOGELIUS
 Gaussian Elimination, Row echelon form

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