In partial pivoting, I encounter something like the following: Let $P_1,...,P_n$ be permutation matrices, and $L$ is a lower, unit triangular matrix corresponding to the subtraction of rows underneath. In a text, the author simply said to $P_1...P_n LP_1^{-1}...P_n^{-1}$ is lower triangular. I cannot understand why it is the case, since if I consider the matrices $L=\begin{bmatrix} 1 & 0\\ -1 & 1 \end{bmatrix}$, $P=\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix}$, the matrix $PLP^{-1}$ turns out to be $\begin{bmatrix} 1 & -1\\ 0 & 1 \end{bmatrix}$ (if I did not calculate it wrong), which is clearly not lower triangular.
Remark: The following is the text I read. The formula (4.15) refers to the formula
$$U=L_{n-1}P_{n-1}...L_1P_1A$$
where $L_i,P_j$ are elementary matrices corresponding to substraction of rows and permutation of rows respectively. The formula comes from the process of partial pivoting.
