If you are given a statement:
"If A has a unique LU-factorization, where L is a lower triangular matrix, then all n leading principal submatrices of A are nonsingular"
How would you show that this is not true?
I wrote a proof like this:
Suppose A is nonsingular with the L U factorization A = LR. Since A is nonsingular it follows that L and R are nonsingular. By definition, we know $A_k=L_k*R_K$. $L_k$ is lower unit triangular and therefore nonsingular. $R_k$ is nonsingular because diagonal entries are amongst the nonzero diagnol entries of R.
This is incorrect. I need to find a counterexample for the beginning statement. But I don't even know where to find a proper countexample. I think the proof I wrote made sense, but perhaps not.