Let $A$ be real symmetric $n\times n$ matrix whose only eigenvalues are $0$ and $1$. Let the dimension of the null space of $A-I$ be $m$. Pick out the true statements.
The characteristic polynomial of $A$ is $(\lambda-1)^m(\lambda)^{m-n}$.
$A^k = A^{k+1}$
The rank of $A$ is $m$.
This is what I did: I found geometric multiplicity corresponding to eigenvalue value $0$ to be $n-m$(using symmetric matrix is diagonalizable ) while geometric multiplicity of eigenvalue value 1 is $m$(that is given) .so the characteristic polynomial of $A$ should be $(\lambda-1)^m(\lambda)^{n-m}$. While I am not sure about other two statements. Any kind of help is highly appreciated. Thanks.