Math 1553 Worksheet: 4.5, 5.15.3
1.
Answer true if the statement is
always
true. Otherwise, answer false.
a)
If
A
is an
n
×
n
matrix and the equation
Ax
=
b
has at least one solution for
each
b
in
R
n
, then the solution is
unique
for each
b
in
R
n
.
b)
Suppose
A
is an
n
×
n
matrix and every vector in
R
n
can be written as a linear
combination of the columns of
A
. Then
A
must be invertible.
c)
Suppose
A
and
B
are invertible
n
×
n
matrices. Then
A
+
B
is invertible and
(
A
+
B
)

1
=
A

1
+
B

1
.
Solution.
a)
True.
The first part says the transformation
T
(
x
) =
Ax
is onto.
Since
A
is
n
×
n
, this is the same as saying
A
is invertible, so
T
is onetoone and onto.
Therefore, the equation
Ax
=
b
has exactly one solution for each
b
in
R
n
.
b)
True. If the columns of
A
span
R
n
, then
A
is invertible by the Invertible Matrix
Theorem. We can also see this directly without quoting the IMT:
If the columns of
A
span
R
n
, then
A
has
n
pivots, so
A
has a pivot in each
row and column, hence its matrix transformation
T
(
x
) =
Ax
is onetoone and
onto and thus invertible. Therefore,
A
is invertible.
c)
False.
A
+
B
might not be invertible in the first place. For example, if
A
=
I
2
and
B
=

I
2
then
A
+
B
=
0 which is not invertible. Even in the case when
A
+
B
is invertible, it still might not be true that
(
A
+
B
)

1
=
A

1
+
B

1
. For
example,
(
I
2
+
I
2
)

1
= (
2
I
2
)

1
=
1
2
I
2
, whereas
(
I
2
)

1
+ (
I
2
)

1
=
I
2
+
I
2
=
2
I
2
.
2.
Find the volume of the parallelepiped naturally formed by
2
1

2
!
,
1
2
1
!
, and
1
3
1
!
.
Solution.
We compute
det
2
1
1
1
2
3

2
1
1
!
=
2det
2
3
1
1

1det
1
3

2
1
+
1det
1
2

2
1
=
2
(
2

3
)

1
(
1
+
6
) +
1
(
1
+
4
)
=

2

7
+
5
=

4.
The volume is
 
4

=
4.