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Row equivalence 31/03

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Row equivalence 31/03

Summary

Let M, M0 be two matrices.

We say that M is row equivalent to M0 if M, M0 have the same size and M0 can be obtained from M by one or more ERO’s.

We indicate this by writing M ∼ M0.

It is obvious that, for any matrices M, M0, M00, we have

• M ∼ M,

• M ∼ M0, M0 ∼ M00 ⇒ M ∼ M00.

But because each type of ERO can be reversed, we also have

• M ∼ M0 ⇒ M0 ∼ M.

These three properties mean that ∼ is an equivalence relation.

Now suppose that M = ( A | B) and M0 = ( A0|B0) represent linear systems with B, B0 column vectors.

Theorem. M ∼ M0 iff the two systems AX = B and A0X = B0 have the same solutions.

If M ∼ M0 then r (M) = r (M0) and r (A) = r (A0) because row reduction preserves the rank of matrices. By (RC1), solutions only exist if all these ranks are all equal.

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