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If A is square matrix such that `A^(2)=A`, then show that `(I+A)^(3)=7A+I`.

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If A is square matrix such that `A^(2)=A`, then show that `(I+A)^(3)=7A+I`.
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Since `A^(2)=A` and `(I+A)=I^(2)+IA+AI+A^(2)`
`=I^(2)+2AI+A^(2)`
`=I+2A+A=I+3A`
and `(I+A).(I+A)(I+A)=I+A)(I+3A)`
`=I+2A+A=I+3A`
and `(I+A).(I+A)(I+A)=(I+A)(I+3A)`
`I^(2)+3AI+AI+3A ^(2)`
`=I+ 4AI+3A`
`=I+7A=7A+I` Hence proved.
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