Can every elementary matrix be written as a product of any number of permutation and diagonal matrices ? That is, given an elementary matrix M can M be written as $M=M_1\cdot M_2\cdots M_k$ such that each $M_i$ is either a permutation matrix or a diagonal matrix ?
Answer
The product of a sequence of permutation and diagonal matrices can have at most one non-zero element in each row and column. As DHMO shows with his example in a comment to your question, row-addition matrices require more than one non-zero element in some column, so those can’t be built from such products.
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