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PROFESSOR KENNETH A. RIBET
Last Midterm Examination
March 18, 2010
2:10–3:30 PM, 10 Evans Hall
Please put away all books, calculators, and other portable electronic devices—anything with an ON/OFF
switch. You may refer to a single 2sided sheet of notes. For numerical questions,
show your work
but do
not worry about simplifying answers. For proofs, write your arguments in complete sentences that explain
what you are doing. Remember that your paper becomes your only representative after the exam is over.
Problem
Possible points
1
6 points
2
12 points
3
6 points
4
6 points
Total:
30 points
1. a.
Use row operations to ﬁnd the inverse of the matrix

2
1
0
4

3
1
1
1

1
.
I’m sure that all of you know how to do this and that most of you will do it correctly. The answer seems to
be
2 1 1
5 2 2
7 3 2
.
b.
Let
A
be an
m
×
n
matrix of rank
m
and let
B
be an
n
×
p
matrix with rank
n
. Determine the rank
of
AB
. Prove that your answer is correct.
Think of
L
A
:
F
n
→
F
m
and
L
B
:
F
p
→
F
n
. Their ranks are equal to the dimensions of the spaces to which
they are mapping. Thus these maps are
onto
. It follows that the same statement is true for
L
A
◦
L
B
=
L
AB
.
In other words,
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This note was uploaded on 07/03/2011 for the course MATH 110 taught by Professor Gurevitch during the Spring '08 term at University of California, Berkeley.
 Spring '08
 GUREVITCH
 Math

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