Property of Adjoint

Basics/adjoint.ipynb

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+    "**1.**\n",
+    "### $A.adj(A)= adj(A).A = |A|I$\n",
+    "<!-- spaces -->\n",
+    "#### Let  A be a square matrix of order n and C represent the cofactor matrix of A and adj(A) represent the adjoint of matrix A .\n",
+    "\n",
+    "$$(adj(A))_{ij} = [C_{ij}]^T = C_{ji}$$\n",
+    "<br>\n",
+    "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.(adj(A))_{rj}$$\n",
+    "<br>\n",
+    "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr}$$\n",
+    "<br> \n",
+    "#### We Know,\n",
+    "<!-- spaces -->\n",
+    "##### Sum of products of  elements with their corresponding cofactor is equal to the determinant of the matrix.\n",
+    "<br>\n",
+    "\n",
+    "$$(A.adj(A))_{ij} = \\sum_{r=1}^{r=n} a_{ir}.C_{jr} = \\left\\{\\begin{array} \\\\ \n",
+    "                                                        |A| & \\mbox{if} \\ i=j \\\\ \n",
+    "                                                         0 & \\mbox{if} \\ i \\neq j\n",
+    "                                                     \\end{array}\n",
+    "\\right.$$\n",
+    "\n",
+    "<br>\n",
+    "\n",
+    "#### So,\n",
+    "<br>\n",
+    "$$ A.adj(A) = \\begin{bmatrix} |A| &  0  &  0  & . & . & 0\n",
+    "                            \\\\  0  & |A| &  0  & . & . & 0\n",
+    "                            \\\\  .  & . & . & . & . & .\n",
+    "                            \\\\  .  & . & . & . & . & .\n",
+    "                            \\\\  0  & 0 & . & . & . & |A|\n",
+    "                            \\end{bmatrix}_{n\\times n}$$\n",
+    "                            \n",
+    "<br>\n",
+    "$$ A.adj(A) = |A|.\\begin{bmatrix} 1 &  0  &  0  & . & . & 0\n",
+    "                            \\\\  0  & 1 &  0  & . & . & 0\n",
+    "                            \\\\  .  & . & . & . & . & .\n",
+    "                            \\\\  .  & . & . & . & . & .\n",
+    "                            \\\\  0  & 0 & . & . & . & 1\n",
+    "                            \\end{bmatrix}_{n\\times n}$$\n",
+    "                            \n",
+    "<br>\n",
+    "$$A.adj(A) = |A|.I_{n}$$\n",
+    "\n",
+    "<!-- spaces -->\n",
+    "##### Similarly we can prove adj(A).A = |A|.I\n",
+    "\n",
+    "\n"
+   ]
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