What is the multiplication matrix for reflecting over the line x=-y?

If you reflect (1,0) over that line you get (0,-1).

If you reflext (0,1) over that line you get (-1,0).

So the matrix implementing this reflection is the 2x2 matrix with (0,-1) as its first column and (-1,0) as its second column.

The same idea works for identifying the matrix of any linear transformation L from the plane to itself. To find the matrix that implements the linear transformation L, it is enough to figure out what L does to (1,0) (and make that the first column of the matrix), and then figure out what L does to (0,1) (and make that the second column of the matrix).