LU Decomposition, Matrices. Find a matrix $A$ that admits NO $LU$ decomposition, even if we only require that $L$ is lower triangular (not necessarily unit) and $U$ is upper triangular. Justify your answer.

QUESTION: Find a matrix $A$ that admits NO $LU$ decomposition, even if we only require that $L$ is lower triangular (not necessarily unit) and $U$ is upper triangular. Justify your answer.