There are three types of elementary row operations: I) row switching, II) row multiplication and III) row addition, corresponding to three kinds of row operation matrix.

My question is that does there exist an elementary operation of one kind, such that it can be decomposed into a sequence of other two operation?

For example, among all row switching operations, does there exist one row switching operation such that it can be decomposed into some row multiplication operations and row addition operations?

From the intuition of handedness, I think this is unlikely to be true.


You indeed can represent row switching as composition of addition and multiplication.

Say you have rows $a, b$. Then following sequence uses only addition and multiplication: $(a, b) \to (-a, b) \to (-a, b - a) \to (a, b - a) \to (b, b - a) \to (b, a - b) \to (b, a)$.

You can't, however, decompose multiplication in addition and switching, as if you start from positive coefficients, they will remain positive after addition and switching, but not necessary after multiplication.

You also can't decompose addition in multiplication and switching: neither multiplication nor switching can increase number of non-zero elements in matrix, but addition can.

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