Is it true that any square matrix $A $ can be factorized as a product of only triangular matrices? That is, can we write $A $ as $\prod_{i=1}^k B_i$, where every $B_i$ is either a lower or an upper triangular matrix (for some natural $ k $)?
Note $ k $ is not assumed to be 2 above. So the question is not (directly) about the $LU$ decomposition.
Comment: We know that for any square matrix $A$ we have: $A=PLU$ where $P$ is a permutation matrix, $L$ is a lower triangular matrix and $U$ is an upper triangular matrix. So the question possibly boils down to whether any permutation matrix can be factorized to triangular matrices.