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.