= Solution
{c}
For any <matrix> $Y$ and compatible <invertible matrix> $P$, the <linear map> $r\mapsto rP$ maps $\operatorname{rowspace}(Y)$ onto $\operatorname{rowspace}(YP)$: a <linear combination> of rows of $Y$ is sent to the same <linear combination> of rows of $YP$. Its <inverse function> is $s\mapsto sP^{-1}$, so these <row spaces> have equal <dimensions>. Hence right <matrix multiplication> by $P$ also preserves <row rank>.
If $X=P^{-1}YP$, <1/a/ii> now gives
$$
\boxed{\operatorname{rowrank}X=\operatorname{rowrank}(YP)=\operatorname{rowrank}Y.}
$$
This proves the required <matrix similarity> invariant without needing to assume equality of <row rank> and <column rank>.
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