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A Solution of Math 115 2002 Midterm Test
Problem 1.
(a) Determine the values of
a
and
b
for which the following system of linear equations has
(i) exactly one solution,
(ii) inFnitely many solution, and
(iii) no solutions.
x
1
+
ax
3
=2
x
1
+
x
2
+(
a
+1)
x
3
=7
x
1
+
x
2
+2
ax
3
=
b
.
(b) ±or the values of
a
and
b
in part (ii) above, give the general solution to the system.
Solution.
(a)
(
A
±
±
b
)
=
⎛
⎝
10
a
11
a
+1
11 2
a
±
±
±
±
±
±
2
7
b
⎞
⎠
R
2
→
R
2
−
R
1
R
3
→
R
3
−
R
1
/
⎛
⎝
a
011
01
a
±
±
±
±
±
±
2
5
b
⎞
⎠
R
3
→
R
3
−
R
2
/
⎛
⎝
a
1
00
a
−
1
±
±
±
±
±
±
2
5
b
−
5
⎞
⎠
Then
(i) the system has exactly one solution when
a
6
=1and
b
∈
R
.
(ii) the system has inFnitely many solutions when
a
b
=5
(iii) the system has no solution when
a
= 1 but
b
6
=5.
(b) When
a
=1,
b
=5,wehave
⎛
⎝
101
000
±
±
±
±
±
±
2
5
0
⎞
⎠
Therefore
x
1
+
x
3
x
2
x
3
.
Let
x
3
=
s
.Th
en
x
1
−
s, x
2
−
s, x
3
=
s
and
⎛
⎝
x
1
x
2
x
3
⎞
⎠
=
⎛
⎝
2
−
s
5
−
s
s
⎞
⎠
=
s
⎛
⎝
−
1
−
1
1
⎞
⎠
+
⎛
⎝
2
5
0
⎞
⎠
.
Problem 2.
(a) A 3
×
3matr
ix
B
has the following elementary row operations performed on it, in the order given:
1)
R
1
→
R
1
+
aR
2
(add
a
t
imesrow2torow1)
2)
R
3
±
R
2
( interchange row 2 and row 3)
3)
R
2
→
bR
2
(multiply row 2 by
b
)
where
a
and
b
are non-zero real numbers.
±ind the matrix
A
such that the matrix product
AB
gives the same result as preforming the
above three elementary row operations on
B
.

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