<ul><li>
 <center>\( {\rm det }(A_1,...,\special{color=red}A_j\special{color=black},\special{color=blue}A_{j+1}\special{color=black}, ..., A_n)=-{\rm det }(A_1,...,\special{color=blue}A_{j+1}\special{color=black},\special{color=red}A_{j}\special{color=black}, ..., A_n) )</center>
<div class="thm"> Echanger deux colonnes conscutives change le signe du dterminant.

\fold{echange1}{<span class="dem">Dmonstration</span>}
</div>
</li><li> S'il existe deux indices \( j ) et \( k ) <span class="definition">diffrents</span>  tels que \( A_j = A_k = B ), <p> <center>\( 
{\rm det }(A_1,...\special{color=red}B\special{color=black}, ..., \special{color=red}B \special{color=black},...,A_n)=0 )</center></p>
<div class="thm"> Si deux colonnes sont gales ou proportionnelles, le dterminant est nul.

\fold{echange2}{<span class="dem">Dmonstration</span>}

</div>
</li><li> si \(k\neq j), 
 <center>\( {\rm det }(A_1,...,\special{color=red}A_j\special{color=black},...,\special{color=blue}A_{k}\special{color=black}, ..., A_n)=-{\rm det }(A_1,...,\special{color=blue}A_{k}\special{color=black},...,\special{color=red}A_{j}\special{color=black}, ..., A_n) )</center>
<div class="thm"> Echanger deux colonnes conscutives change le signe du dterminant.

\fold{echange3}{<span class="dem">Dmonstration</span>}
</div>
</li>

<li>si \( j\neq m ) 
 <center>\( {\rm det }(A_1,...,\special{color=red}A_j\special{color=black}, ...,\special{color=blue}A_m\special{color=black},..., A_n)= {\rm det }(A_1,...,\special{color=red}A_j\special{color=black}+\special{color=blue}A_m\special{color=black}, ...,\special{color=blue}A_m\special{color=black},..., A_n) )
</center>
 <div class="thm"> On ne change pas le dterminant 
en ajoutant  une colonne une combinaison linaire des 
<span class="definition">autres</span>  colonnes.


\fold{echange4}{<span class="dem">Dmonstration</span>}

</div>
</li>
<li> Si \( \special{color=red}B \special{color=black}= \sum_{i=1}^n a_i A_i =a_1A_1+...+ \special{color=red}a_mA_m\special{color=black}+... + a_nA_n)
<center>\(\special{color=red} a_m\special{color=black}{\rm det }(A_1,...,\special{color=red}A_m\special{color=black},..., A_n)= 
{\rm det }(A_1,...,\special{color=red}B\special{color=black},..., A_n) )</center> o \( B ) est mis  la place de la \( m )-ime colonne. 


\fold{echange5}{<span class="dem">Dmonstration</span>}
</li></ul>