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What are the three elementary lower triangular $4 \times 4$ matrices and what does their operation do? How can I prove that for all of these, $\det(L)=1$ and $L(x)^{-1}=L(-x)$?

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"The" three elementary lower...matrices? Over what field? What is L? How do you define "elementary" matrix: both transvections and permutation matrices of the unit one, or only permutations? What is L? How do you define "lower triangular: the main diagonal entries must be zero and everything below it, or only below it must be zero? What is L? Is it raining in London right now? (dismiss the last question, but all the others are important info you must add, imo)...oh, and what is L, BTW? –  DonAntonio Feb 24 '13 at 12:57
    
I have the feeling $L_i(x)$ represents the elementary row operation $r_i \rightarrow r_i + x r_{i-1}$ which works for $i = 2, 3, 4$ –  muzzlator Feb 24 '13 at 13:48

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Let $L_i(x)$ represent the elementary row operation $r_i \rightarrow r_i + x r_{i-1}$

$$ L_2(x) = \left( \begin{array}{cccc}1 & 0 & 0 & 0 \\ x & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) $$

$$ L_3(x) = \left( \begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & x & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right) $$

$$ L_4(x) = \left( \begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & x & 1 \end{array} \right) $$

Then various ways to prove $\det(L_i(x)) = 1$, depending on which definition of $\det$ you know. As for proving $L(x)^{-1} = L(-x)$, just multiply each of these by $L_i(-x)$ or point to the row operation and show that it cancels out.

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