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MATH 225
ASSIGNMENT 1
Work the following problems.
These are not to be handed in.
In the problems below,
R
denotes a ring,
Z
denotes the set of integers,
and
R
denotes the set of real numbers.
§2.1 p. 117 #1 – 7; pp. 118 – 119 #1 – 15, 26, 27.
A. Let
A
∈
M
n
(
Z
) have all its entries zero except for one 1 each row, one 1 in each column, and
at most one 1 on the diagonal.
Give examples of such matrices for
n
= 2, 3, 4, 5.
In general, show that for any matrix
A
with this property,
A
T
also satisfies this property.
B. A general symmetric 2
×
2 matrix has the form:
a
b
b
a
"
#
$
%
'
.
Write out a general symmetric 4
×
4 matrix and general skew-symmetric 4
×
4 matrix.
C. Show that for any matrix
A
, (
A
T
)
T
=
A
.
If
B
∈
R
7x9
must
B
=
A
T
for some matrix
A
?
Why, in which case describe
A
, or why not?
D. For any given
A
∈
M
2
(
R
) let
B
∈
M
2
(
R
) be defined by
B
ij
=
A
ij
2
.
If tr(
A
) = tr(
B
) = 0 show that
A
11
=
A
22
= 0.
Does this remain true if instead
B
ij
=
A

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