Linear Algebra
Mathematics 124 B Spring 2007
Problem Assignment # 2
Due Wednesday, February 21
1) Section 2.3: 44, 53
2) Section 2.4: 35, 38, 46, 72, 76
3) Section 3.1: 6, 10 12, 37, 38
4) Let O be the n
๎
n
zero
matrix
.
If
A
is
a
nonzero
n
๎
n matrix and p is a positive integer greater than 1, then we
say that A is nilpotent of index p if A
p
๎
O
,
but
A
k
๎
O
for
1
๎
k
๎
p.
Show that C๎
๎
๎
๎
๎
๎
๎
๎
๎
๎
๎
0
1
2
0 0 4
0 0 0
๎
๎
๎
๎
๎
๎
๎
๎
๎
๎
is nilpotent of index 3.
5) Can a nilpotent matrix of index p be invertible? Justify your answer.
6) Suppose that A
๎
B
are
n
๎
n
matrices
such
that
A
3
๎
B
3
and
A
2
๎
B
๎
B
2
๎
A. Can A
2
๎
B
2
be invertible?
Hint: Consider the product
๎
A
2
๎
B
2
๎
๎
๎
A
๎
B
๎
.
m124bhw2.nb
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