Math 136
Assignment 8 Solutions
1.
For each of the following matrices, ±nd the inverse, or show that the matrix is not invertible.
a)
A
=
1

12
315
223
.
Solution: To determine if
A
is invertible we write [
A

I
] and row reduce:
1

12100
3
1
5 0 1 0
2
2
3 0 0 1
∼
1

1
2
1
0
0
04

1

310
0
0
0
1

11
Since the RREF of
A
will not be
I
, it follows that
A
is not invertible.
b)
B
=
1

102
0
1
1 0
2

235
1
0
1 3
.
Solution: To determine if
E
is invertible we write [
E

I
] and row reduce:
1

1021000
0
1
1 0 0 1 0 0
2

2350010
1
0
1 3 0 0 0 1
∼
1000
10
3
8
3

1
3

5
3
0100
1
3
2
3

1
3
1
3
0010

1
3
1
3
1
3

1
3
0001

1

1
0
1
.
Since the RREF of
E
is
I
, it follows that
E
is invertible and
E

1
=
10
3
8
3

1
3

5
3
1
3
2
3

1
3
1
3

1
3
1
3
1
3

1
3

1

1
0
1
2.
Let
B
=
2

0
1
1
1

1

1
. Find
B

1
and use it to solve
B±x
=
±
d
, where
±
d
= (4
,

2
,
3).
Solution: To ±nd the inverse of
B
we write [
B

I
] and row reduce:
2

1
1
1 0 0
0
1
1
0 1 0
1

1

1001
∼
1 0 0
0
1
1
010

1
/
23
/
21
0 0 1
1
/
2

1
/
2

1
Therefore
B

1
=
0
1
1

1
/
/
1
/
2

1
/
2

1
.
We have
±x
=
B

1
(
B±x
)=
B

1
4

2
3
=
0
1
1

1
/
/
1
/
2

1
/
2

1
4

2
3
=
1

2
0
.
1
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3.
a) Prove that if
A
and
B
are
n
×
n
matrices such that
AB
is invertible, then
A
and
B
are
invertible.
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 Winter '08
 All
 Matrices, E3 E2 E1

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