Covariance Matrix Question

I think I'm reading the information matrix wrong, and thus am getting the delta method calculations wrong as well.



Suppose we have a two-variable function g(x,y) and we are estimating Var(g(x,y)) via the delta method. If we are given the 2x2 information matrix (rows, columns: 11, 12, 21, 22), my understanding was that the variance of x is 11, variance of y is 22, and covariance of x and y is 12 = 21.



Apparently I'm incorrect about 12 and 21. Why do we need to calculate the inverse of the information matrix, and what exactly are the values in cells 12 and 21 if they are not the covariances?



Thanks.

Covariance Matrix Question