So I can find two ways to reduce the matrix $A$ into echelon (i.e. upper triangular) form:
1) I can factorize the matrix as $A=LU$, where $L$ is lower triangular and $U$ is upper triangular. This is probably the first factorization that gets taught in a linear algebra class (it is deeply connected to the the echelon form of the matrix).
2) Alternately, I can factorize the matrix as $A=QR$, where $Q$ is orthonormal or unitary (depends if your matrices are over the real or the complex numbers) and $R$ is right (i.e. upper) triangular.
In general, these are completely different factorizations. To convince yourself for the 2 x 2 case, I'd suggest something like this:
1) Start with the matrix
2) Multiply $A$ out and figure out its LU factorization
3) Compare the LU factorization to the original expression for $A$