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MATH 115 SOLUTION SET 7
ANSWERS TO SELECTED PROBLEMS
§
5.1
7. The point of this problem is to extend the deﬁnition of determinant from
matrices to linear transformations. The result is always that a property satisﬁed
by matrices is always satisﬁed by linear transformations as well:
a. If
Q
is the change of basis matrix from
β
to
γ
, then [
T
]
β
=
Q
[
T
]
γ
Q

1
, so that
[
T
]
β
and [
T
]
γ
are similar and therefore have the same determinant.
b.
T
is invertible if and only if the matrix [
T
]
β
is invertible, which is so if and
only if det[
T
]
β
6
= 0. But det[
T
]
β
is the same as det
T
.
c. We have det(
T

1
) = det[
T

1
]
β
= det[
T
]

1
β
= (det
T
)

1
.
d. We have det(
TU
) = det([
TU
]
β
) = det([
T
]
β
[
U
]
β
) = det([
T
]
β
)det([
U
]
β
) =
det(
TU
).
e. We have det(
T

λI
V
) = det([
T

λI
V
]
β
) = det([
T
]

λI
). The second equality
comes from Thm. 2.8.
/
8. a.
T
is invertible if and only if det
T
6
= 0, which is true if and only if 0 is not
a root of the characteristic polynomial det(
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