MATH1111/201011/
Tutorial II
1
MATH1111 Tutorial II
1. Let
A
and
B
be two matrices satisfying
AB
=
BA
.
(a) Must
A
and
B
be square matrices of the same order? Explain.
(b) Show that if the order is odd, then
A
or
B
is singular.
(c) Must
A
or
B
be singular if the order is even? Justify your answer.
2. Let
V
be a vector space and
x
2
V
. A student says that it is
ambiguous
to write
2
x
, because
one may interpret
2
x
in di erent ways, such as
(2
x
)
;
(
1)(2
x
)
;
(
2)
x
;
2(
x
)
;
(2
(
1))
x
;
What is your comment? Give your justi cation.
3. Are the following sets vector spaces with the indicated operations? If not, why not?
(a) The set
V
of all polynomials of exact degree 3 together with 0; operations are the usual
operations on polynomials.
(b) The set
V
of 2
2 matrices whose determinant equal zero; operations are usual matrix
operations.
(c) The set
V
of 2
2 matrices whose column sums are equal; operations are usual matrix
operations.
(d) The set
V
=
f
1
g
consisiting of only the element 1 (
2
R
) where both 1 + 1 and
1 (for
any
2
R
) are de ned to be 1.
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 Spring '10
 Dr,Li

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