if A(2 1 3)
3 4 5
B=(4 3)
0 6
5 7
find A+B' , A'+B and verity (A+B')'=A'+B
Answers
Answer:
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Class 12
>>Maths
>>Matrices
>>Transpose of a Matrix
>>For the matrices A and B , verify that
Question
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For the matrices A and B, verify that (AB)
′
=B
′
A
′
where
(i) A=
⎣
⎢
⎢
⎡
1
−4
3
⎦
⎥
⎥
⎤
,B=[
−1
2
1
]
(ii) A=
⎣
⎢
⎢
⎡
0
1
2
⎦
⎥
⎥
⎤
,B=[
1
5
7
]
Medium
Solution
verified
Verified by Toppr
(i) AB=
⎣
⎢
⎢
⎡
1
−4
3
⎦
⎥
⎥
⎤
[
−1
2
1
]=
⎣
⎢
⎢
⎡
−1
4
−3
2
−8
6
1
−4
3
⎦
⎥
⎥
⎤
∴(AB)
′
=
⎣
⎢
⎢
⎡
−1
2
1
4
−8
−4
−3
6
3
⎦
⎥
⎥
⎤
Now, A
′
=[
1
−4
3
],B
′
=
⎣
⎢
⎢
⎡
−1
2
1
⎦
⎥
⎥
⎤
∴B
′
A
′
=
⎣
⎢
⎢
⎡
−1
2
1
⎦
⎥
⎥
⎤
[
1
−4
3
]=
⎣
⎢
⎢
⎡
−1
2
1
4
−8
−4
−3
6
3
⎦
⎥
⎥
⎤
Hence, we have verified that (AB)
′
=B
′
A
′
(ii) AB=
⎣
⎢
⎢
⎡
0
1
2
⎦
⎥
⎥
⎤
[
1
5
7
]=
⎣
⎢
⎢
⎡
0
1
2
0
5
10
0
7
14
⎦
⎥
⎥
⎤
∴(AB)
′
=
⎣
⎢
⎢
⎡
0
0
0
1
5
7
2
10
14
⎦
⎥
⎥
⎤
Now, A
′
=[
0
1
2
],B
′
=
⎣
⎢
⎢
⎡
1
5
7
⎦
⎥
⎥
⎤
∴B
′
A
′
=
⎣
⎢
⎢
⎡
1
5
7
⎦
⎥
⎥
⎤
[
0
1
2
]=
⎣
⎢
⎢
⎡
0
0
0
1
5
7
2
10
14
⎦
⎥
⎥
⎤
Hence, we have proved that (AB)
′
=B
′
A
′