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If Z is an idempotent matrix, then (I+Z)...

If Z is an idempotent matrix, then `(I+Z)^(n)`

A

`I+2^(n)Z`

B

`I+(2^(n)-1) Z`

C

`I-(2^(n)-1)Z`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
Z is idempotent, then
`Z^(2)=Zimplies Z^(3), Z^(4), ..., Z^(n)=Z`
`:. (I+Z)^(n)=^(n)C_(0)I^(n)+ .^(n)C_(1)I^(n-1) Z+.^(n)C_(2)I^(n-2)Z^(2)+ +^(n)C_(n)Z^(n)`
`=^(n)C_(0)I+ .^(n)C_(1)Z+ .^(n)C_(2)Z+ .^(n)C_(3)Z+ +^(n) C_(n)Z`
`=I +(.^(n)C_(1)+.^(n)C_(2)+.^(n)C_(3)+ +.^(n)C_(n))Z`
`=I+(2^(n)-1)Z`
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