Linear Algebra with Applications (3rd Edition)

110.201 Linear Algebra
1st Quiz, Solution
February 15, 2005
Problem 1 Given the following system of equations:
x - 3 y + z = 1
x + y + 2 z = 14
find all solutions using Gauss-Jordan elimination procedure. Is this an example
of consistent system? Why?
Solution
1 - 3 1
1 1 2
1
14
Row(II)-Row(I):
1 - 3 1
0 4 1
1
13
Row(II) ÷ 4:
1 - 3 1
0 1
1
4
1
13
4
Row (I)+ Row (II) × 3:
1 0
7
4
0 1
1
4
43
4
13
4
1

Therefore, the answer is:



x =
43
4
-
7
4
z
y =
13
4
-
1
4
z
z = any real number
Of course, consistent system and has infinite solutions.
Problem 2 Find the rank of the following matrix




1 0 1 1 2
- 1 1 1 0 0
0 1 1 1 1
1 0 1 1 2




Solution




1 0 1 1 2
- 1 1 1 0 0
0 1 1 1 1
1 0 1 1 2




+(I)
-(I)
then:




1 0 1 1 2
0 1 2 1 2
0 1 1 1 1
0 0 0 0 0




-(II)
then:




1 0 1 1 2
0 1 2 1 2
0 0 - 1 0 - 1
0 0 0 0 0




× ( - 1)
then:




1 0 1 1 2
0 1 2 1 2
0 0 1 0 1
0 0 0 0 0




- (III)
- (2 × (III))
2

then:




1 0 0 1 1
0 1 0 1 0
0 0 1 0 1
0 0 0 0 0




Here, we see, the reduced row-echelon form has 3 leading 1’s, therefore, the rank
is 3.