Show that the characteristics roots of an idempotent matris are either 0 or 1

Explanation

Since A is an idempotent matrix, we have
`A^(2)=A`
Let X be a latent vector of the matrix A corresponding to the latent root `lambda` so that
`AX=lambdaX` (1)
or `(A-lambda I)X=O`
such that
`X ne O`
On pre-multiplying Eq. (1) by A, we get
`A(AX)=A(lambdaX)=lambda(AX)`
i.e., `(A A)X=lambda(AX)`
or `AX=lambda^(2) X`
or `lambdaX=lambda^(2)X" "( :' A^(2)=A)`
or `(lambda^(2)-lambda)X=O" "( :' AX=lambdaX)`
or `lambda^(2)-lambda=0`
or `lambda(lambda-1)=0" "( :' X ne 0)`
`implies lambda=0, lambda=1`