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If B is an idempotent matrix, and A=I-B,...

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`

Text Solution

Verified by Experts

The correct Answer is:
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`.
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