If A and B are matrices, then which from the following is true A : A+ B.+B+ A/ B : (AT)T + A /С: AB=BA /D: ABBA Questions Bookmark Invalid
Answers
Answer:
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Answer:
Correct options are A) , B) and C)
Given, AB=A and BA=B
Take, AB=A
⇒A(BA)=A [ Since, B=BA ]
⇒(AB)A=A
⇒AA=A [ Since, AB=A ]
⇒A
2
=A
Therefore, A is an idempotent matrix.
Take, BA=B
⇒B(AB)=B [ Since, A=AB ]
⇒(BA)B=B
⇒BB=B [ Since, BA=B ]
⇒B
2
=B
Therefore, B is an idempotent matrix.
Now, A=AB
Applying transpose on both sides, we get
A
T
=(AB)
T
⇒A
T
=B
T
.A
T
....(i)
Also, B=BA
Applying transpose on both sides, we get
B
T
=(BA)
T
⇒B
T
=A
T
.B
T
...(ii)
From equation (i) and equation (ii), we get
(A
T
)
2
=A
T
and
(B
T
)
2
+B
T
Therefore, A
T
and B
T
are alos idempotent matrices.
Hence, the correct options are (A),(B) and (C).