How to prove that LU decomposition is unique?

Question

given the following matrix how could I prove that LU decomposition of it is unique?

A=
1 3 1
2 9 2
1 3 1

L=
1 0 0
2 1 0
1 0 1

U=
1 3 1
0 3 0
0 0 0

Answer 1

To prove it is easy. Suppose that $$L_{1} U_{1}=L_{2} U_{2}$$ where $L_{1}, L_{2}$ are unit lower triangular and $U_{1}, U_{2}$ are upper triangular and invertible. (They must be invertible if $L_{i} U_{i}$ is so.) We now rearrange the equation to get $$L_{2}^{-1} L_{1}=U_{2} U_{1}^{-1}$$ Now the left side is unit lower triangular and the right side is upper triangular, but only the identity matrix satisies both conditions at once. Therefore we have $$L_{2}^{-1} L_{1}=I=U_{2} U_{1}^{-1}$$ hence $$L_{1}=L_{2}, U_{1}=U_{2}$$ proving the desired uniqueness.