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QUIZ # 2–MATH349
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1. Find bases for
W
and
W
⊥
⊂
R
3
, where
W
=
x
y
z
:
x
+
y

2
z
= 0
.
2. If
x
is an eigenvector of
A
with eigenvalue
λ
= 3, show that
x
is
also an eigenvector of
A
2

5
A
+3
I
. What is the corresponding
eigenvalue ?
3. Let
A
be a square matrix such that that
A
3
=
A
. Show that
λ
=

1
,λ
= 0 and
λ
= 1 are the only possible eigenvalues.
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Unformatted text preview: 4 Prove that if A and B are similar matrices, then A is invertible if and only if B is invertible. 5 Let A,B be n × n orthogonal matrices. Show that A ( A T + B T ) B = A + B and use the identity to prove that if det( A ) + det( B ) = 0, then A + B is not invertible....
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This note was uploaded on 02/20/2011 for the course MATH 302 taught by Professor Staff during the Fall '08 term at University of Delaware.
 Fall '08
 Staff
 Math

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