linear algebra - A symmetric matrix with eigenvalues all $0$ or all $1$: does it equal $0$ or identity?

I have these general wondering about matrices but I don't know to proceed with a proof or a counter example. Suppose that $A$ (dimension $n\times n$) is a real symmetric matrix.

1. If $A$ has $n$ eigenvalues that are all $1$'s, does $A$ equal the identity matrix?


2. If $A$ has $n$ eigenvalues that are all $0$'s, does $A$ equal the zero matrix?

Can someone elucidate things for me please?

Edit: I learned/can look up diagonalization theorems for real matrices.

Answer

The answer to both is yes.

Hint: All symmetric matrices are diagonalizable. That is, $A$ is similar to a diagonal matrix with the eigenvalues of $A$ on the diagonal.