I'm afraid you're completely on the wrong track. is a matrix, is a number (it's the dot product). To convince yourself of this, look at the sizes of the vectors (n x 1) against (1 x n), along with the definition of matrix multiplication and the size of the resulting multiplication.
So, is the matrix . What's in the denominator is a number: the dot product.
I would ask yourself what the definitions of "idempotent" and "symmetric" are. What are those?
Break the problem down into 2 parts. First, try to show that the matrix is symmetric (i.e. transpose(A) = A). Then, try to show that the matrix is idempotent (A^2 = A). Since you have shown both separately, it means both are true.
Hint: You do not need to expand the vector x into its components to do this problem.