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.
