# Fail to understand elementary matrices properties in matrix LU factorization.

I am reading about LU factorization for the first time. I encounter a theorem that says that if we can obtain a Upper-triangular row equal matrix U from matrix A with elementary matrices call them E1, E2, .., Ek that are not row interchanged then we can have LU factorization. The reason for not interchanging rows as said is to keep Ei matrices Lower-Triangular. I think if We never interchange rows but for example Add s times of row 2 to row 1 We contract the Lower-triangularity of Ei matrices too, but this operator didn't forbid in the theorem as you see.

Thank You in Advance.

You are right that in order to ensure that the elementary operations define a lower triangular matrix, one can only use those that add a multiple of a row to a row below it, or that multiply a row by a nonzero scalar (the latter if you allow diagonal entries of $L$ that differ from $1$). The point is that, provided you always find a nonzero coefficient on the diagonal, you never need any other operations to transform the matrix into an upper triangular matrix. (In cases where at some point you find a $0$ on the diagonal with some nonzero coefficient below it, no $LU$ decomposition exists at all.)