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MATH227 Quiz 5 (Fall 2006)
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1. Suppose
A
is an
n
×
n
matrix and there exist
n
×
n
matrices
C
and
D
such that
CA
=
I
n
and
AD
=
I
n
. Prove that
C
=
D
. Is
A
invertible? Why?
(5 points)
Note that
C
=
C
(
I
n
) =
C
(
AD
) = (
CA
)
D
= (
I
n
)
D
=
D.
Thus,
C
=
D
. Now by the deﬁnition, since
CA
=
I
n
and
AC
=
AD
=
I
n
,
A
is
invertible.
2. Suppose
A
,
B
, and
C
are invertible
n
×
n
matrices. Show that
ABC
is also invertible
and ﬁnd its inverse.
(5 points)
As
A
,
B
and
C
are invertible,
A

1
,
B

1
and
C

1
exist. Let
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Unformatted text preview: D = C1 B1 A1 . Then D ( ABC ) = C1 B1 ( A1 A ) BC = C1 B1 ( I n ) BC = C1 ( B1 B ) C = C1 ( I n ) C = C1 C = I n and ( ABC ) D = AB ( CC1 ) B1 A1 = AB ( I n ) B1 A1 = A ( BB1 ) A1 = A ( I n ) A1 = AA1 = I n . i.e., D ( ABC ) = I n = ( ABC ) D . Thus, ABC is invertible and D = C1 B1 A1 is the inverse. 1...
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This note was uploaded on 04/23/2008 for the course MATH 227 taught by Professor Sze during the Fall '06 term at UConn.
 Fall '06
 Sze
 Matrices

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