when do you ever sum over all the entries in a matrix?

Imagine you have $n$ measurements of a quantity $y$. What is your best estimate of the value of $y$? It turns out that if you have an estimate for the covariance matrix of $y$, the information in (expected inverse variance from) your $n$ data points is given by the sum of the entries of the inverse of that covariance matrix. This fact is obvious in retrospect, but also confused me, since this is such a non-coordinate-free thing to do to a matrix!