show that the characterstic roots of an idempotent matrix are either z

1.	show that the characterstic roots of an idempotent matrix are either zero or unity.
Answer» Let A be an idempotent matrix, then `A^(2)=A` If `lambda` be an eigen value of the matrix A correspondin to eigen vector X,so that `AX=lambdaX` where `X!=0` from Eq. (ii) , `A(AX)=A(lambdaX)` `rArr (A A)X=lambda(AX)` ` rArr A^(2)x=lambda (lambdaX)` `rArr AX = lambda^(2)X` `rArr lambdaX=lambda^(2)X` `rArr (lambda-lambda^(2))X=0` `rArr lambda-lambda^(2)=0` `therefore lambda=0` or lambda=1`

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show that the characterstic roots of an idempotent matrix are either zero or unity.

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