I'm afraid you're completely on the wrong track. \(\displaystyle xx'\) is a matrix, \(\displaystyle x'x\) is a number (it's the dot product). To convince yourself of this, look at the sizes of the vectors \(\displaystyle x\) (n x 1) against \(\displaystyle x'\) (1 x n), along with the definition of matrix multiplication and the size of the resulting multiplication.

So, \(\displaystyle A\) is the matrix \(\displaystyle \frac{1}{x'x}\,xx'\). 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?