<?xml version="1.0"?>
<div><span class="kw">theorem </span><a NAME="T36"><span class="comment"><font color="firebrick">:: LAPLACE:36</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">K</font> being   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a><br/>  for <font color="Olive" title="b3">M</font> being   <a href="matrix_1.html#NM2" title="MATRIX_1:NM.2">Matrix</a> of <font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">K</font>  st <font color="Olive" title="b3">M</font> is  <a href="matrix_6.html#V1" title="MATRIX_6:attr.1">invertible</a>  holds <br/> for <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_0.html#K2" title="MATRIX_0:func.2">Indices</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b3">M</font> <a href="matrix_6.html#K1" title="MATRIX_6:func.1">~</a></span>)</span> <a href="matrix_0.html#K3" title="MATRIX_0:func.3">*</a> (<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="matrix_3.html#K12" title="MATRIX_3:func.12">Det</a> <font color="Olive" title="b3">M</font></span>)</span> <a href="algstr_0.html#K9" title="ALGSTR_0:func.9">"</a></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Olive" title="b2">K</font></span>)</span> <a href="laplace.html#K1" title="LAPLACE:func.1">.</a> (<span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p4">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Olive" title="b2">K</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><font color="Olive" title="b4">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Olive" title="b5">j</font></span>)</span>)</span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="laplace.html#K5" title="LAPLACE:func.5">Minor</a> (<font color="Olive" title="b3">M</font>,<font color="Olive" title="b5">j</font>,<font color="Olive" title="b4">i</font>)</span>)</span></div></div>
