Multiplication of two Gaussians

I know that the multiplication of two Gaussian densities is another Gaussian (but not normalized). For example:
N(a,A)*N(b,B) is proportional to N(c,C).

The moments of the new distribution can be calculated (using matrix notation) as
C = [A^(-1) + B^(-1)]^(-1)
c = CA^(-1)a + CB^(-1)b

Can this be shown analytically by integrating the product of two normal densities? Offhand I can't see how one would arrive at this...

For example the expectation of a gaussian density with mean u1 and variance s1^2 is u*s1^2, so the expectation of a product would be:
u1*s1^2*u2*s2^2

which differs from the matrix notation result that is a weighted sum and not a product...
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