3. Let . Find the matrix such that
A B AB (AB)−1 =B−1A−1
A=(1 2 1 0 1 0 1 3 2)
Answers
Answer:
Given A=[
3
2
7
5
] and B=[
6
7
8
9
].
Inverse of AB :
AB=[
3
2
7
5
][
6
7
8
9
]
⇒AB=[
18+49
12+35
24+63
16+45
]
⇒AB=[
67
47
87
61
]
Now, ∣AB∣=4087−4089=−2
Since, ∣AB∣
=0
Hence, (AB)
−1
exists.
(AB)
−1
=
∣AB∣
adj(AB)
Now, we will find adj(AB)
For this , we will find co-factors of each element of AB.
C
11
=(−1)
1+1
61=61
C
12
=(−1)
1+2
47=−47
C
21
=(−1)
2+1
87=−87
C
22
=(−1)
1+1
67=67
Hence, the cofactor matrix is [
61
−87
−47
67
]
adjAB=C
T
=[
61
−47
−87
67
]
⇒(AB)
−1
=
∣AB∣
adj(AB)
=
−2
1
[
61
−47
−87
67
]
Inverse of A :
We have A=[
3
2
7
5
]
∣A∣=15−14=1
Since, ∣A∣
=0
Hence, A
−1
exists.
A
−1
=
∣A∣
adjA
Now, we will find adjA
For this , we will find co-factors of each element of A.
C
11
=(−1)
1+1
5=5
C
12
=(−1)
1+2
2=−2
C
21
=(−1)
2+1
7=−7
C
22
=(−1)
1+1
3=3
Hence, the cofactor matrix is [
5
−7
−2
3
]
adjA=C
T
=[
5
−2
−7
3
]
⇒A
−1
=
∣A∣
adjA
=[
5
−2
−7
3
]
Inverse of B :
We have A=[
6
7
8
9
]
∣B∣=54−56=−2
Since, ∣B∣
=0
Hence, B
−1
exists.
B
−1
=
∣B∣
adjB
Now, we will find adjB
For this , we will find co-factors of each element of B.
C
11
=(−1)
1+1
9=9
C
12
=(−1)
1+2
7=−7
C
21
=(−1)
2+1
8=−8
C
22
=(−1)
1+1
6=6
Hence, the cofactor matrix is [
9
−8
−7
6
]
adjB=C
T
=[
9
−7
−8
6
]
⇒B
−1
=
∣B∣
adjB
=
−2
1
[
9
−7
−8
6
]
Now, B
−1
A
−1
=
−2
1
[
9
−7
−8
6
][
5
−2
−7
3
]
=
−2
1
[
45+16
−35−12
−63−24
49+18
]
⇒B
−1
A
−1
=
−2
1
[
61
−47
−87
67
]
Hence, (AB)
−1
=B
−1
A
−1