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If A B=Aa n dB A=B , then which of the f...

If `A B=Aa n dB A=B ,` then which of the following is/are true? `A` is idempotent b. `B` is idempotent c. `A^T` is idempotent d. none of these

A

(a) A is idempotent

B

(b) B is idempotent

C

(c) `A^T` is idempotent

D

(d) none of these

Text Solution

Verified by Experts

The correct Answer is:
A, B, C
Given, `AB=A, BA=B`
`implies BxxAB=BxxA`
or `(BA)B=B`
or `B^(2)=B`
Also,
`AxxBA=AB`
`implies (AB)A=A`
`implies A^(2)=A`
Now `(A^(T))^(2)=(A^(T)xxA^(T))=(AxxA)^(T)=(A^(2))^(T)=A^(T)`
Similarly, `(B^(T))^(2)=B^(T)`
Hence, `A^(T)` and `B^(T)` are idempotent.
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