Math 136
Assignment 8 Solutions
1.
For each of the following matrices, find the inverse, or show that the matrix is not
invertible.
a)
A
=
2
1
4
4
Solution: We have
A

1
=
1
2(4)

1(4)
4

1

4
2
=
1
4
4

1

4
2
.
b)
B
=
1
5

3
1
3
1

1

4
1
Solution: Row reducing [
B

I
] gives
1
5

3
1
0
0
1
3
1
0
1
0

1

4
1
0
0
1
∼
1
5

3
1
0
0
0

2
4

1
1
0
0
0
0
3
/
2
1
/
2
1
Since the RREF of
B
is not
I
,
B
is not invertible.
2.
Let
A
=
1
2
3
1
2
4
1

1
0
and
B
=
1
2
3
6
2
2

1
1
0
.
a) Find det
A
and det
B
.
Solution: We have
det
A
=
1
2
3
1
2
4
1

1
0
=
1
3
3
1
3
4
1
0
0
= 3
det
B
=
1
2
3
6
2
2

1
1
0
=
3
2
3
8
2
2
0
1
0
= 18
b) Find det(
AB
).
Solution: det(
AB
) = det
A
det
B
= 54
c) Find det(
A
+
B
).
Solution: det(
A
+
B
) =
2
4
6
7
4
6
0
0
0
= 0
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
d) Find
A

1
.
Solution: We have
1
2
3
1
0
0
1
2
4
0
1
0
1

1
0
0
0
1
∼
1
0
0
4
/
3

1
2
/
3
0
1
0
4
/
3

1

1
/
3
0
0
1

1
1
0
Thus
A

1
=
4
/
3

1
2
/
3
4
/
3

1

1
/
3

1
1
0
.
3.
Write each of the following matrices and their inverses as a product of elementary
matrices.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '08
 All
 Math, Matrices

Click to edit the document details