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Math 136  Final Exam Winter 2009
NOTE:
The questions on this exam does not exactly refect which questions will be on this terms
exam. That is, some questions asked on this exam may not be asked on our exam and there may
be some questions on our exam not asked here.
1.
Short Answer Problems
a) Find the inverse o±
A
=
±
12

13
²
.
b) Let
L
1
,L
2
:
R
3
→
R
3
where
L
1
is a refection in the
x
2
x
3
plane and
L
2
is a refection
in the
x
1
x
2
plane. Find the matrices ±or
L
1
,
L
2
and
L
2
◦
L
1
.
c) Show that i±
C
is a 5
×
5 matrix with real entries, then
C
has at least one real eigenvector.
d) Let
A
be an
n
×
n
matrix such that
A
2
= 0. Prove that the columnspace o±
A
is
a subset o± the nullspace o±
A
.
2.
Let
A
=
10

56
20
.
a) Find the reduced row echelon ±orm
B
o±
A
.
b) Write
A
as a product o± elementary matrices and
B
.
3.
Let
C
be a square matrix whose nullspace is
t
1
1
1
1
³
³
³
³
t
∈
R
.
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This note was uploaded on 04/30/2010 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.
 Winter '08
 All
 Math

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