**Problem**. Let be a field. Let be two matrices with entries in Suppose that and . Prove that

**Solution**. Let and supppose that are the corresponding linear transformations defined by and respectively. Let and Note that, since and we have by the rank-nullity theorem and thus As a result Also, since we have Therefore, by the rank-nullity theorem, and hence

Now let Then and therefore i.e. Hence and so Thus

Finally, and the rank-nullity theorem give us

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I was looking for this, thanks! 🙂