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APPENDIX D
SOLUTIONS TO PROBLEMS
D.1
(i)
016
21
7
2
0
61
2
180
4
5
0
5
36
24
300
⎛⎞
−−
⎛
⎜⎟
==
⎜
⎝⎠
⎝
AB
⎞
⎟
⎠
(ii)
BA
does not exist because
B
is 3
×
3 and
A
is 2
×
3.
D.3
Using the basic rules for transpose,
(
)()
()
′
′′
′
′
′
=
=
XX
X X
, which is what we wanted to
show.
D.5
(i) The
n
×
n
matrix
C
is the inverse of
AB
if and only if
C
(
AB
) =
I
n
and (
AB
)
C
=
I
n
.
We
verify both of these equalities for
C
=
B
1
A
1
.
First, (
B
1
A
1
)(
AB
) =
B
1
(
A
1
A
)
B = B
1
I
n
B
=
B
1
B
=
I
n
.
Similarly, (
AB
)(
B
1
A
1
) =
A
(
BB
1
)
A
1
=
AI
n
A
1
=
AA
1
=
I
n
.
(ii) (
ABC
)
1
= (
BC
)
1
A
1
=
C
1
B
1
A
1
.
D.7
We must show that, for any
n
×
1 vector
x
,
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This note was uploaded on 05/03/2009 for the course ECON 418 taught by Professor Breman during the Spring '08 term at University of Arizona Tucson.
 Spring '08
 BREMAN
 Econometrics

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