## Similar matrices have the same row and column ranks

Let and be similar square matrices over a field . Prove that and have the same row and column ranks.

Since and are similar, there exists a linear transformation and two bases and (of , where is the dimension of and ) such that and . In the previous exercise we saw that .

Recall that the transpose of an matrix , denoted , is the matrix .

Lemma: If is a commutative ring, an matrix over , and a matrix over , then . Proof: We have .

Clearly and . Moreover, if , then . In particular, if is invertible, then so is , and .

Now for some invertible matrix . Thus . By the above argument, and have the same column rank. Thus and have the same row rank.

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