1
MAT1332 Assignment #5 solutions
Question 1
Determine the solution of the following systems :
a
)
x
+ 2
y
+
z
=
2
x
+ 3
y
+ 2
z
=
5
2
x
+ 9
y
+ 7
z
=
1
Solution :
The augmented matrix associated to this system is
1
2
1
2
1
3
2
5
2
9
7
1
.
Replace the second row R2 with R2

R1, and replace the third row R3 with R3

2R1
(2R1 means 2 times row 1) :
1
2
1
2
0
1
1
3
0
5
5

3
.
Replace the third row R3 with R3

5R2 :
1
2
1
2
0
1
1
3
0
0
0

15
.
The bottom row, when expressed as equations, says 0 =

15. Since that never holds,
the original system of equations has no solution.
b
)
2
x
+ 2
y
+
z
=
6
x
+ 3
y
+ 2
z
=
18
2
x
+ 2
y
+ 7
z
=
12
Solution :
The augmented matrix associated to this system is
2
2
1
6
1
3
2
18
2
2
7
12
.
For convenience, we switch the first two rows R1
↔
R2. This way the leading entry in
the first row is a 1 instead of a 2. (It is also correct to avoid this step.)
1
3
2
18
2
2
1
6
2
2
7
12
.
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Replace R2 with R2

2R1 and R3 with R3

2R1, to clear our the rest of the third
column :
1
3
2
18
0

4

3

30
0

4
3

24
.
The leading entry in the second row is in the second column. We next clear out the entries
below it, by replacing R3 with R3

R2 :
1
3
2
18
0

4

3

30
0
0
6
6
.
Our matrix is now in rowechelon form. We now continue through to reduced rowechelon
form. We first make all leading entries equal to one, by scaling each row. We leave R1
alone, multiply R2 by

1
/
4, and multiply R3 by 1
/
6, to get :
1
3
2
18
0
1
3
/
4
30
/
4
0
0
1
1
.
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 Spring '07
 MUNTEANU
 Row echelon form, rowechelon form

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