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Math 20F
First Midterm
January 27, 2003
•
PRINT NAME
•
Write version on your blue book and
VERSION B
hand in this exam inside your blue book.
•
There are a total of 40 points possible.
•
No BOOKS, NOTES or CALCULATORS are allowed.
•
You must show your work to receive credit.
1. (10 pts.) Prove or give a counterexample:
(a) If
A
and
B
are 2
×
2 matrices, then
AB
=
BA
.
(b) For every matrix matrix
A
, the ﬁrst entry in
A
T
A
is nonnegative.
(In other words, if
B
=
A
T
A
, then
b
11
≥
0.)
2. (15 pts.) Let
A
=
11

10
22 3 0
10 0

1
00 1 0
.
(a) How many solutions does
A
x
=(0100)
T
have? Justify your answer.
(b) How many solutions does
A
x
=(0010)
T
have? Justify your answer.
(c) Either ﬁnd a
b
so that
A
x
=
b
has exactly one solution or explain why this is
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 Winter '07
 Enright
 Math, Linear Algebra, Algebra

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