Math 136
Assignment 9 Solutions
1.
Calculate the determinant of the following matrices.
a)
A
=
01

10
3 0
0
2
0 1
2
1
5 0
0
7
Solution: Expand along the fourth row:
det
A
=
±
±
±
±
±
±
±
±

3 0
0
2
0 1
2
1
5 0
0
7
±
±
±
±
±
±
±
±
=

3
±
±
±
±
±
±
1

121
007
±
±
±
±
±
±

5
±
±
±
±
±
±
1

002
±
±
±
±
±
±
=(

3)(7)
±
±
±
±
1

1
12
±
±
±
±

(5)(

2)
±
±
±
±
1

1
±
±
±
±
=

21(2 + 1) + 10(2 + 1) =

63 + 30 =

33
b)
B
=
1

1
5
5
2
3
1
2
4
6

1

3
8
0

2
1
1
2

11

1
1
0
Solution: Since adding a multiple of one row to another and adding a multiple of one column
to another does not change the determinant, we have
det
B
=
±
±
±
±
±
±
±
±
±
±
1

1
5
5
2
3
1
2
4
6

1

3
8
0

2
1
1
2


1
1
0
±
±
±
±
±
±
±
±
±
±
=
±
±
±
±
±
±
±
±
±
±
1

1
5
5
2
04

13

11
0
0

4
13
5
0
02

3

6

1
03

6

4

2
±
±
±
±
±
±
±
±
±
±
=
±
±
±
±
±
±
±
±
4

13

11
0

4
13
5
0
2

3

6

1
3

6

4

2
±
±
±
±
±
±
±
±
=6
±
±
±
±
±
±
4

13
0
2

3

1
3

6

2
±
±
±
±
±
±
±
±
±
±
±
±
4

13
0
00

1


2
±
±
±
±
±
±
±
±
±
±
4

13

±
±
±
±
= 6(

13) =

78
c)
C
=
a
b
c
a
+
b
2
bc
+
b
2
2
2
Solution: Since adding a multiple of one row to another does not change the determinant, we
have
det
C
=
±
±
±
±
±
±
a
b
c
a
+
b
2
+
b
2
2
2
±
±
±
±
±
±
=
±
±
±
±
±
±
a b c
b b b
2 2 2
±
±
±
±
±
±
=
±
±
±
±
±
±
a b c
b b b
0 0 0
±
±
±
±
±
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 Winter '08
 All
 Determinant, Matrices, Invertible matrix, Det

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