# Elementary Lower Matrices

First of all forgive me for my lack of format.

I want to prove that the following elementary lower triangular $nxn$ matrix $Li(x)= I-xe(i)^T$ where $x=[0 \ldots 0 x(i+1) \ldots x(n)]^T$ has the properties:

det $Li(x)=1$,

$Li(x)^{-1}=Li(-x)$,

and pre-multiplying a matrix $A=[a_{ij}]$ for $i,j$ $\in [1:n]$ by $Li(x)$ leaves the first $i$ rows unchanged and from row $(i+1)$ subtracts the row vector $\displaystyle x(k)[a_{i,1} a_{i,2} \ldots a_{i,n}]$ from row $k$ of $A$.

Thanks :)

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