MAS 3105
May 14, 2009
Quiz 2 and Key
Prof. S. Hudson
1) If possible, find a nonzero 2x2 matrix
A
such that
A
2
=
O
(the zero matrix). If it is
not possible, explain why not. Note:
nonzero
means at least one entry of
A
is not zero.
2) Write
v
as a linear combination of
u
and
w
. For maximum credit, solve this using a
reliable method (guessing the answer may only get partial credit).
v
=
1
2
u
=
2
1
w
=
1
1
3) Choose ONE of these to prove. You can answer on the back.
a) If
A
is a symmetric nonsingular matrix, then
A

1
is also symmetric [if you know a
formula for the transpose of
A

1
, you can use it without proving it].
b) Prove this part of the TFAE theorem: If
A
is row equivalent to
I
, then
A
is nonsingular.
Remarks and Answers:
Average 44/60. A’s = 5060, B’s 4449, C’s 3843, D’s 3237.
1) There are many examples (all singular, of course), but several people used this one:
A
=
0
1
0
0
2) [first version] There were two versions of the quiz. The first one shows the correct year,
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 Spring '09
 JULIANEDWARDS
 Linear Algebra, Matrices, upper right corner, Prof. S. Hudson, symmetric nonsingular matrix

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