Somewhat challenging linear algebra proof

[workerant]

Homework Statement

Let LU and L'U' be two LU decompositions for an invertible matrix. Prove L=L' and U=U', thus the LU decomposition for an invertible matrix is unique.

Homework Equations

The Attempt at a Solution

I honestly do not really know what to do. I suppose I could consider something with the diagonals being equal and try to show that the entries would be the same for the identity matrix, but I'm not sure how to approach any of this.

 

[Mark44]

Homework Statement

Let LU and L'U' be two LU decompositions for an invertible matrix. Prove L=L' and U=U', thus the LU decomposition for an invertible matrix is unique.

The Attempt at a Solution

I honestly do not really know what to do. I suppose I could consider something with the diagonals being equal and try to show that the entries would be the same for the identity matrix, but I'm not sure how to approach any of this.

What you're trying to prove isn't true in general. Here's a counterexample.
----A

| 3 -6 |
| -2 5 |

Note that A is invertible because its determinant is nonzero.

Here's one decomposition:
----L-----------U

| 3   0|  |1  -2 |
| -2  1|  |0   1 |

It's pretty easy to check that LU = A, so the matrices above constitute an LU factorization of A.
Here's another decomposition of A:
----L'-----------U'

| 1      0|  |3  -6 |
| -2/3  1|  |0   1 |

It's also easy to check that L'U' = A, so here is a different LU factorization of A.
(Sorry this is so clunky. I'm not up on writing matrices in LaTeX.)

What do these decompositions say about the statement you're trying to prove?