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Math 20F
First Midterm Solutions
January 27,2003
The two versions are nearly the same and problems 1 and 3 have been interchanged.
The solutions here are for version A with notes on changes for B.
1. (#3 in version B) (a)
AB
=
BA
=
I
. I mentioned in class that
AB
=
I
or
=
I
is
suﬃcient, so either “
AB
=
I
” and “
=
I
” are also acceptable.
(b) transpose
2. There are many ways to convert a matrix to row echelon form. I’ll choose one way.
Rn means row n. For version A:
10

00 0
1
11 0

1
22 0
3
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
00
01
→


1
1
5
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
add (

2)
×
(R3) to R4
then switch R2 and R3
→

01 1

1
1
0
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ

subtract R1 from R2
add (

5)
×
(R3) to R4
Thus (a) has solutions; e.g., (1

100)
T
; however (b) does not have a solution because
the last row of the row echelon form augmented matrix is inconsistent.
For version B:
11

22 3
0
10 0

1
00 1
0
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
→

5
0
0


1
1
0
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
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This note was uploaded on 04/16/2008 for the course MATH 20F taught by Professor Buss during the Spring '03 term at UCSD.
 Spring '03
 BUSS
 Math

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