Do you know how to calculate an eigenvalue? It involves a polynomial. If you don't, check your book. If you do, try to do it and write back.
On b), when you calculate an eigenvalue, you're setting the determinant equal to zero, right? We're looking for a matrix that has non-trivial solutions. Make sure you follow this. If you don't, write back. Now if the eigenvalue is zero, then ... the determinant is already equal to zero. Try it with a 2x2 matrix. Same thing works with larger matrices.