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Solutions to Homework 9.
Math 110, Fall 2006.
Prob 4.3.10.
Since det(
AB
) = det(
A
) det(
B
) for any two square matrices of the same order, this implies,
by induction, that (det
M
)
k
= det(
M
k
) for all
k
∈
IN. Since the determinant of the zero matrix is zero, this
shows that, for a nilpotent matrix
M
, (det
M
)
k
= 0, hence det
M
= 0.
Prob 4.3.11.
By Theorem 4.8, det
M
t
= det
M
. Now, if
M
is skewsymmetric, then
det
M
= det
M
t
= det(

M
) = (

1)
n
det
M,
the last equality by the
n
linearity of the determinant. If
n
is odd, this gives det
M
=

det
M
, i.e.,
det
M
= 0. If
n
is even, it does not imply anything.
Prob 4.3.14.
Since
β
consists of
n
vectors,
β
is a basis if and only if these vectors are linearly independent,
which is equivalent to the map
L
B
being onetoone. Since the matrix
B
is square, this is in turn equivalent
to
B
being invertible, hence having a nonzero determinant. Thus,
β
is a basis if and only if det
B
6
= 0.
Prob 4.3.15.
If matrices
A
and
B
are similar, then, for some invertible matrix
Q
, we have
A
=
QBQ

1
.
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This note was uploaded on 03/06/2008 for the course MATH 110 taught by Professor Valdimarsson during the Spring '08 term at UCLA.
 Spring '08
 VALDIMARSSON
 Determinant, Matrices

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