Question

If B is an idempotent matrix, and `A=I-B`, then
A. `A^(2)=A`
B. `A^(2)=I`
C. `AB=O`
D. `BA=O`

Answer 1

Correct Answer - A::C::D
B is an idempotent matrix
`:. B^(2)=B`
Now, `A^(2)=(I-B)^(2)`
`=(I-B) (I-B)`
`=I-IB-IB+B^(2)`
`=I-B-B+B^(2)`
`=I-2B+B^(2)`
`=I-2B+B`
`=I-B`
`=A`
Therefore, A is idempotent. Again,
`AB=(I-B)B=IB-B^(2)=B-B^(2)=B^(2)=B^(2)-B^(2)=O`
Similarly, `BA=B(I-B)=BI-B^(2)=B-B=O`.