If AB=A and BA=Bm then which of the following is/are true ?

Options

A. A is idempotent
B. B is idempotent
C. `A^T` is idempotent
D. none of these

Answer

A, B, C

Explanation

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.