# LU decomposition implies Gaussian Elimination without pivoting

If Gaussian elimination can be carried out without pivoting for A, then A has an LU decomposition. Is the converse true: if A has an LU decomposition, then Gaussian elimination can be carried out (in exact arithmetic, disregarding stability issues) without pivoting? If yes, can somebody explain carefully why this works in the other direction as well? Im new to numerical methods, thanks.