MATH111 Additional Problems: Solution
The problems are quite difficult if your goal is just a pass for the midterm.
1. A
n
×
n
matrix
A
is called
upper triangular
if
a
ij
= 0 whenever
i > j
. Prove that the product of
two upper triangular matrices is upper triangular.
Soln:
Let
A
= (
a
ij
),
B
(
b
ij
) be two upper triangular matrices. Then when
i > j
,
(
AB
)
ij
=
X
k
a
ik
b
kj
=
X
k
≤
j
a
ik
b
kj
+
X
k>j
a
ik
b
kj
.
Since
i > j
, for
k
≤
j
,
a
ik
= 0. So the first term equals 0. For
k > j
,
b
kj
= 0. So the second term
equals 0.
So (
AB
)
ij
= 0 whenever
i > j
, which means
AB
is upper triangular.
2. A square matrix
A
is called
nilpotent
if
A
k
= 0 for some
k >
0. Prove that if
A
is nilpotent, then
I
+
A
is invertible.
Soln:
First we let
k
be the smallest integer such that
A
k
= 0. We want to find
D
such that
D
(
I
+
A
) = (
I
+
A
)
D
=
I
. By guessing the form of
D
, we treat
A
as a real number and we do long
division. Then we guess
D
=
I

A
+
A
2

A
3
+
· · ·
=
I

A
+
A
2

A
3
+
· · ·
+ (

1)
k

1
A
k

1
since
A
k
= 0.
Then we try (
I
+
A
)
D
= (
I
+
A
)(
I

A
+
· · ·
+ (

1)
k

i
A
k

1
=
I
and
D
(
I
+
A
) = (
I
+
A
)(
I

A
+
· · ·
+ (

1)
k

i
A
k

1
=
I
.
3. The
trace
of a square matrix is the sum of its diagonal entries:
tr
A
=
a
11
+
a
22
+
· · ·
+
a
nn
.
a) Show that tr(
A
+
B
) = tr
A
+ tr
B
, and that tr
AB
= tr
BA
.
b) Show that if
B
is invertible, then tr
A
= tr
BAB

1
.
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 Spring '07
 Cheng
 Linear Algebra, Algebra, Determinant, Addition, Matrices, Diagonal matrix, Triangular matrix, soln

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