Compute the Second Order Derivative of Digital Image with Finite Differences
Pay attention that convolution mean flipping the kernel both on the x and y axis.
Hence the first element is a multiplication of $ -1 \cdot -1 $ which yields $ 1 $ as in the answer in the original post.
Pay attention that this is a discrete approximation of the gradient based on Finite Differences. This specific one is based on the Sobel 1st derivative filter.
In Finite Differences Coefficients you may find more variations.
As usual, the longer the filter the better the accuracy of the approximation with longer tarnsients.