diff --git "a/Chapitre 7 - Le d\303\251terminant d'une matrice/7.3 Crit\303\250re d'inversibilit\303\251.ipynb" "b/Chapitre 7 - Le d\303\251terminant d'une matrice/7.3 Crit\303\250re d'inversibilit\303\251.ipynb"
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+++ "b/Chapitre 7 - Le d\303\251terminant d'une matrice/7.3 Crit\303\250re d'inversibilit\303\251.ipynb"	
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+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "import numpy as np\n",
+    "import sympy as sp\n",
+    "import sys, os \n",
+    "#sys.path.append('../Librairie')\n",
+    "#from AL_Fct import *\n",
+    "from IPython.display import display, Latex\n",
+    "def whether_invertible(A):\n",
+    "    \"\"\"Judge whether the matrix A is invertible by calculating the determinant.\"\"\"\n",
+    "    A_RREF = A.rref()[0]\n",
+    "    A_det = sp.Float(A.det(), 4)\n",
+    "    detA_s = sp.latex(A)\n",
+    "    detAr_s = sp.latex(A_RREF)\n",
+    "    Ar_det = sp.Float(A_RREF.det(), 4)\n",
+    "    display(Latex(\"$\" +   \"\\det A\"+\"=\"+ \"\\det\" + detA_s + \"=\" + \"k \\cdot\"+ \"\\det\" + detAr_s+  \"= k \\cdot\"+ \"{}\".format(Ar_det)+\"=\" + \"{}\".format(A_det)+\"$\" ))\n",
+    "    display(Latex(\"Where $k$ is a non-zero constant. \"))\n",
+    "    if detA_s == 0:\n",
+    "        display(Latex(\"$\\det A$ equals to zero so the matrix A is is singular.\" ))\n",
+    "    else:\n",
+    "        display(Latex(\"$\\det A $ is not equal to zero, so the matrix A is invertible.\"))\n",
+    "\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### PROPOSITION :\n",
+    "\n",
+    "#### Si $A=(a_{ij})\\in M_{n\\times n}(\\mathbb{R})$. Alors $A$ est inversible si et seulement si $\\det A \\neq 0$.\n",
+    "\n",
+    "\n",
+    "This theorem  adds the statement “$\\det A \\neq 0$” to the Invertible Matrix Theorem. A useful corollary is that $\\det A = 0$ when the columns of $A$ are linearly dependent. Also, $\\det A = 0$ when the rows of $A$ are linearly dependent. (Rows of $A$ are columns of $A^T$, and linearly dependent columns of $A^T$ make $A^T$ singular. When $A^T$ is singular, so is $A$, by the Invertible Matrix Theorem.) In practice, linear dependence is obvious when two columns or two rows are the same or a column or a row is zero.\n",
+    "\n",
+    " \n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "#### Exemple :  Compute det A, where $A = \\begin{pmatrix}\n",
+    "3 & -1 &  2  & -5\\\\\n",
+    "0 & 5 &-3 & -6\\\\\n",
+    "-6 & 7 & -7 & 4\\\\\n",
+    "-5 & -8 & 0 & 9\n",
+    "\\end{pmatrix}$\n",
+    "\n",
+    "\n",
+    "**SOLUTION** Add 2 times row 1 to row 3\n",
+    "\n",
+    "\n",
+    "\n",
+    "$$ \\det A = \\det \\begin{pmatrix}\n",
+    "3 & -1 &  2  & -5\\\\\n",
+    "0 & 5 &-3 & -6\\\\\n",
+    "0 & 5 &-3 & -6\\\\\n",
+    "-5 & -8 & 0 & 9\n",
+    "\\end{pmatrix} = 0 $$ \n",
+    "\n",
+    "$\\det A = 0$ and the matrix $A$ is is singular, because the second and third rows of the matrix are equal.\n",
+    "#### Exercise :  Judge whether the matrix A is invertible by calculating its determinant.\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/latex": [
+       "$\\det A=\\det\\left[\\begin{matrix}1 & 2 & 0.75\\\\0.5 & 2.0 & 4\\\\1 & 6 & 43\\end{matrix}\\right]=k \\cdot\\det\\left[\\begin{matrix}1 & 0 & 0\\\\0 & 1 & 0\\\\0 & 0 & 1\\end{matrix}\\right]= k \\cdot1.000=27.75$"
+      ],
+      "text/plain": [
+       "<IPython.core.display.Latex object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "Where $k$ is a non-zero constant. "
+      ],
+      "text/plain": [
+       "<IPython.core.display.Latex object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    },
+    {
+     "data": {
+      "text/latex": [
+       "$\\det A $ is not equal to zero, so the matrix A is invertible."
+      ],
+      "text/plain": [
+       "<IPython.core.display.Latex object>"
+      ]
+     },
+     "metadata": {},
+     "output_type": "display_data"
+    }
+   ],
+   "source": [
+    "A = sp.Matrix([[1,2,3/4], [0.5,20/10,4], [1,6,43]])\n",
+    "whether_invertible(A)\n"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.7.4"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}