Solve for parameters of a system of non linear equations that have forms of $0 = e^x + A_1 e^x + A_2 e^{2x} + A_3$

Question

I'm self studying math and came across a problem that I have to solve for parameters of the following equations, $$ 0 = e^{-x} - A_1 - A_2 e^{-x} - A_3e^{-2x} \\ 0 = e^{-2x} - A_1e^{-x} - A_2 - A_3e^{-x} \\ 0 = e^{-3x} - A_1e^{-2x} - A_2e^{-x} - A_3 \\ $$ The answer given is $A_1=e^{-x}, A_2=0, A_3 = 0$, but there's no method shown for finding these. I was thinking I can put them in matrix form but I don't think that works, any suggestion of systemically finding the solution besides substituting individually?