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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be   <a href="matrix_1.html#NM3" title="MATRIX_1:NM.3">Matrix</a> of <font color="Maroon" title="c1">n</font>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:90"/>
<font color="Maroon" title="c2">A</font> is <a href="matrixr2.html#V1" title="MATRIXR2:attr.1">invertible</a>
 ;<br/>

<a NAME="E2:90"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrixr2.html#K4" title="MATRIXR2:func.4">1_Rmatrix</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="matrixr2.html#D6" target="_self" title="MATRIXR2:def.6">AA444</a></i>;<br/>
<a NAME="E3:90"/><b>then </b>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">A</font> <a href="matrixr2.html#K1" title="MATRIXR2:func.1">*</a> <span class="p2">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrixr2.html#T72" target="_self" title="MATRIXR2:th.72">AA220</a></i>;<br/>
<a NAME="E4:90"/><b>then </b><i><font color="Green" title="E61">A2</font></i>: 
<span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p2">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="matrixr2.html#T46" target="_self" title="MATRIXR2:th.46">AA219</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:90_1"/>
(  <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0 or  <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_1"><b>suppose </b></a><a NAME="E1:90_1_1"/>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_1"/>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> 
 <b>by </b><i><a class="txt" href="matrixr2.html#E4:90"><i><font color="Green" title="E61">A2</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="90_1_2"><b>suppose </b></a><a NAME="E1:90_1_2"/>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:90_1_2"/>
 <a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K7" title="MATRIXR2:func.7">Inv</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="matrixr2.html#K3" title="MATRIXR2:func.3">Det</a> <font color="Maroon" title="c2">A</font></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a> 
 <b>by </b><i><a class="txt" href="matrixr2.html#E4:90"><i><font color="Green" title="E61">A2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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