A Frobeniusmatrix is the matrix with the following properties: $e_k=(0,...,0,1,0,....,0)^T$ is the $k$th base vector of $\mathbb{R}^n$. And $l_k\in\mathbb{R^n}$ is a vector such that $(l_k)_i=0$, for all $i\leq k$. The $n\times n$ matrix $L_k:=I-l_k\cdot e_k^T$ is defined as Frobeniusmatrix.
For $r\geq k\geq n$ let $P_r\in \{0,1\}^{n\times n}$ be a permuataionmatrix such that $P_re_i=e_i$ for all $i\leq r$. Show that
$$P_rL_k=\bar{L_k}P_r$$
for all $k\in \{1,...,n\}$ where $\bar{L_{k}}= I-P_rl_ke_k^T$ is defined over the corresponding vector $l_k$ from $L_k$
$P_rL_k=\bar{L_k}P_r$
$P_r(I-l_k\cdot e_k^T)=(I-P_rl_ke_k^T)P_r$
$P_r-P_r(l_k\cdot e_k^T)=P_r-P_rl_ke_k^TP_r$
$P_r(l_k\cdot e_k^T)=P_rl_ke_k^TP_r$
$(l_k\cdot e_k^T)=(l_ke_k^T)P_r$
Can someone help me with the last step? Or give me a clue please?