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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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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