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If A is an idempotent matrix, then show ...

If A is an idempotent matrix, then show that B=l-A is also idempotent and AB=BA=0

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As A is idempotent `A^(2)=A`
`B^(2)=(i-A)^(2)=(l-A)(l-A)-lA-Al+A A`
`=l-A-A+A^(2)=l-2A+A^(2)=l-2A+A=l-A=B`
Thus B is idempotent
`AB=A(l-A)=A-A^(2)=A-A=0`
`andBA=(l-A)A=A-A^(2)=A-A=0`
Hence `AB=0=BA`
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