Math 2940, Prelim 1 Solutions
Fall 2011
1) A matrix
A
was reduced to
A
0
=
1 0
2
0 1

1
0 0
0
by the following row operations in the
given order:
2 times Row 1 subtracted from Row 2, 4 times Row 1 added to Row 3, Row 2 multiplied
by

1
2
and ﬁnally Row 2 subtracted from Row 3.
(a) (10 points) What was the original matrix
A
?
(b) (8 points) What matrix multiplication combines all the row operations, in other words
for what matrix
L
is
A
0
=
LA
?
Solution:
(a) Working backwards, we add Row 2 to Row 3 to get
1 0
2
0 1

1
0 1

1
then
multiply Row 2 by

2 to get
1
0
2
0

2
2
0
1

1
subtract 4 times Row 1 from Row 3 to get
1
0
2
0

2
2

4
1

9
and add 2 times Row 1 to Row 4 to get
A
=
1
0
2
2

2
6

4
1

9
.
(b) We have
1
0 0
0
1 0
0

1 1
1
0 0
0

1
/
2 0
0
0 1
1 0 0
0 1 0
4 0 1
1 0 0

2 1 0
0 0 1
A
=
1 0
2
0 1

1
0 0
0
so
L
=
1
0 0
0
1 0
0

1 1
1
0 0
0

1
/
2 0
0
0 1
1 0 0
0 1 0
4 0 1
1 0 0

2 1 0
0 0 1
=
1
0 0
0

1
/
2 0
0
1
/
2 1
1 0 0

2 1 0
4 0 1
=
1
0 0
1

1
/
2 0
3
1
/
2 1
.
2) (16 points) Find all
~x
that simultaneously satisfy both of the following two systems.
1 2 1
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 '05
 HUI
 Math, Linear Algebra, Algebra, Row, additivity

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