An $LU$-factorization of a matrix $A$ is a way to write $A$ as the product of a lower triangular matrix $L$ and an upper triangular matrix $U$, where the lower triangular matrix has $1$'s on its diagonal. Prove that the following statement is equivalent to a nonsingular $n \times n$ matrix $A$ having an $LU$-factorization:
A zero pivot position does not emerge during row-reduction to upper triangular form with Type III operations (i.e. operations of the form $R_k\mapsto R_k - cR_j$ for rows $R_k, R_j$).
I think zero pivots can only occur if there are only zeroes in the entries below the main diagonal in a column when doing Gaussian elimination. I've seen a proof online that a matrix possessing an $LU$-factorization is equivalent to each leading principal submatrix of $A$ being nonsingular (i.e. each square submatrix with a top-left corner at $A$'s top left corner). This proof could be useful. Also, I know that the matrices $L$ and $U$ are nonsingular and $L$ has unit diagonal entries. I'm not sure if a proof by contradiction is useful.