{"id":36929,"date":"2022-11-22T18:24:23","date_gmt":"2022-11-22T12:54:23","guid":{"rendered":"https:\/\/yanamtakshashila.com\/?p=36929"},"modified":"2022-11-25T20:10:09","modified_gmt":"2022-11-25T14:40:09","slug":"algebra-formulae","status":"publish","type":"post","link":"https:\/\/yanamtakshashila.com\/?p=36929","title":{"rendered":"UNIT &#8211; I ALGEBRA (Formulae)"},"content":{"rendered":"\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {MATRICES\\ and\\ DETERMINANTS}}\\]<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/hooks.min.js?ver=dd5603f07f9220ed27f1\" id=\"wp-hooks-js\"><\/script>\n<script src=\"https:\/\/yanamtakshashila.com\/wp-includes\/js\/dist\/i18n.min.js?ver=c26c3dc7bed366793375\" id=\"wp-i18n-js\"><\/script>\n<script id=\"wp-i18n-js-after\">\nwp.i18n.setLocaleData( { 'text direction\\u0004ltr': [ 'ltr' ] } );\n\/\/# sourceURL=wp-i18n-js-after\n<\/script>\n<script  async src=\"https:\/\/cdnjs.cloudflare.com\/ajax\/libs\/mathjax\/2.7.7\/MathJax.js?config=TeX-MML-AM_CHTML\" id=\"mathjax-js\"><\/script>\n<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Determinant\\ of\\ Second\\ order:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =\\begin{vmatrix}\na_1 &amp; b_1 \\\\\na_2 &amp; b_2 \\\\\n\\end{vmatrix}\\ = a_1b_2\\ -\\ a_2b_1\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Determinant\\ of\\ Third\\ order:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\begin{vmatrix}\na_1 &amp; a_2 &amp; a_3 \\\\\nb_1 &amp; b_2 &amp; b_3 \\\\\nc_1 &amp; c_2 &amp; c_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =a_1\\begin{vmatrix}\nb_2 &amp; b_3 \\\\\nc_2 &amp; c_3 \\\\\n\\end{vmatrix}\\ -\\ a_2\\begin{vmatrix}\nb_1 &amp; b_3 \\\\\nc_1 &amp; c_3 \\\\\n\\end{vmatrix}\\ +\\ a_3\\begin{vmatrix}\nb_1 &amp; b_2 \\\\\nc_1 &amp; c_2 \\\\\n\\end{vmatrix}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\Delta =a_1(b_2c_3\\ -\\ b_3c_2)\\ &#8211; a_2 (b_1c_3\\ -\\ b_3c_1) + a_3(b_1c_2\\ -\\ b_2c_1)\\ \n\\hspace{10cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {Singular\\ and\\ Non-Singular\\ Matrix}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<p>A square matrix A is called a singular matrix<\/p>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[if\\ \\begin{vmatrix} A \\\\ \\end{vmatrix}\\ = 0\\ and\\ non\\ \u2013\\ singular\\ matrix\\ if\\ \\begin{vmatrix} A \\\\ \\end{vmatrix}\\ \\neq {0}\\ \\hspace{10cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:inline-block;width:300px;height:250px\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"5990643685\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {APPLICATION\\ OF\\ MATRICES\\ and\\ DETERMINANTS}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Minor\\ of\\ an\\ element\\ of\\ a\\  Matrix}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Minor\\ of\\ an\\ element\\ is\\ a\\ determinant\\ obtained\\ by\\ deleting\\ the\\ row\\ and\\ column\\ in\\ which\\ the\\ element\\ occurs\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Cofactor\\ of\\ an\\ element of\\ a\\  Matrix}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Cofactor\\ of\\ an\\ element\\ is\\ a\\ signed\\ minor\\ of\\ that\\ element\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\therefore\\ cofactor\\ of\\ a_{ij} = (-1)^{i\\ +\\ j}\\ minor\\ of\\ a_{ij}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple}{Method\\ for\\ to\\ find\\ adjoint\\ of\\ Matrix\\ of\\ order\\ 3\\ (order 2)}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[i)\\ A\\ is\\ square\\ Matrix\\ of\\ order\\ 3\\ (order 2)\\ \\hspace{5cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[ii)\\ Find\\ the\\ co-factor\\ of\\ all\\ the\\ elements\\ of\\ det\\ A\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[iii)\\ Form\\ the\\ matrix\\ by\\ replacing\\ all\\ the\\ elements\\ of\\ A\\ by\\ the\\ corresponding\\ cofactor\\ in\\ \\begin{vmatrix}\nA \\\\\n\\end{vmatrix}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[iv)\\ Then\\ take\\ the\\ Transpose\\ of\\ that\\ matrix,\\ then\\ we\\ get\\ adj. A.\\ \\hspace{8cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Inverse\\ of\\ Matrix}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {green} {\\boxed {A^{-1} = \\frac{1}{\\begin{vmatrix} A \\\\ \\end{vmatrix}}\\ adj.\\ A}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {Rank\\ of\\ Matrix:}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Let\\ A\\ be\\ any\\ m\u00d7n\\  matrix.\\ The\\ order\\ of\\ the\\ largest\\ square\\ sub\\ matrix\\ of\\ A\\ whose\\ determinant\\]\\[ has\\ a\\ non\\ -\\ zero\\ value\\ is\\ known\\ as\\ the\\ rank\\ of\\ the\\ matrix\\ A\\]\\[and\\ is\\ denoted\\ by\\ \\rho(A)\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {purple} {SOLUTION\\ OF\\ SIMULTANEOUS\\ EQUATIONS\\ USING\\ CRAMERS\\ RULE}\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a_1x\\ + b_1y\\ +\\ c_1z\\ = d_1\\ &#8212;&#8211; (1)\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a_2x\\ + b_2y\\ +\\ c_2z\\ = d_2\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[a_3x\\ + b_3y\\ +\\ c_3z\\ = d_3\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {black}{Solution:}\\ to\\ find\\ x,\\ y,\\ z\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Step\\ 1:\\ \\Delta = \\begin{vmatrix}\na_1 &amp; b_1 &amp; c_1 \\\\\na_2 &amp; b_2 &amp; c_2 \\\\\na_3 &amp; b_3 &amp; c_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Step\\ 2:\\ \\Delta_x = \\begin{vmatrix}\nd_1 &amp; b_1 &amp; c_1 \\\\\nd_2 &amp; b_2 &amp; c_2 \\\\\nd_3 &amp; b_3 &amp; c_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Step\\ 3:\\ \\Delta_y= \\begin{vmatrix}\na_1 &amp; d_1 &amp; c_1 \\\\\na_2 &amp;  d_2 &amp;  c_2 \\\\\na_3 &amp; d_3 &amp; c_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Step\\ 4:\\ \\Delta_z = \\begin{vmatrix}\na_1 &amp; b_1 &amp; d_1 \\\\\na_2 &amp; b_2 &amp; d_2 \\\\\na_3 &amp; b_3 &amp; d_3 \\\\\n\\end{vmatrix}\\ \\hspace{20cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[Solution\\ is\\ x=\\ \\frac{\\Delta_x}{\\Delta}.\\ y=\\ \\frac{\\Delta_y}{\\Delta},\\ z=\\ \\frac{\\Delta_z}{\\Delta}\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<script async=\"\" src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\" crossorigin=\"anonymous\"><\/script>\n<ins class=\"adsbygoogle\" style=\"display:block\" data-ad-format=\"autorelaxed\" data-ad-client=\"ca-pub-9453835310745500\" data-ad-slot=\"4869133702\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\LARGE{\\color {purple} {BINOMIAL\\ THEOREM}}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Binomial\\ theorem\\ for\\ a\\ positive\\ integral\\ index}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ \u2018n\u2019\\  is\\ any\\ positive\\ integer,\\  then\\  \\hspace{15cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(X\\ +\\ a)^n\\ =\\ X^n\\ +\\ nc_1 X^{n-1}a\\ +\\ \\ nc_2 X^{n-2}a^2\\ +\\ &#8230;&#8230;&#8230;&#8230;&#8230;&#8230;&#8230;..\\ +\\ \\ nc_r X^{n-r}a^r\\ +\\ &#8230;&#8230;.\\ a^n\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {General\\ term\\ of\\ the\\ expansion\\ of\\ (X\\ +\\ a)^n}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[T_{r + 1} = nC_rx^{n-r} a^r.\\ \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {To\\ find\\ the\\ Middle\\ term\\ of\\ the\\ expansion\\ of\\ (X\\ +\\ a)^n}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {brown} {cases}:\\ \\hspace{18cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[1.\\ If\\ n\\ is\\ an\\ even\\ number,\\ there\\ is\\ one\\ middle\\ term\\ =\\ (\\frac{n+2}{2})^{th}\\ term\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[2.\\ If\\ n\\ is\\ odd\\ number,\\ there\\ are\\ two\\ middle\\ terms\\ (\\frac{n+1}{2})^{th}\\ and\\ \\ (\\frac{n+1}{2})^{th}\\ term\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[\\color {royalblue} {Binomial\\ theorem\\ for\\ rational\\ index}:\\ \\hspace{20cm}\\] <\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[If\\ \u2018n\u2019\\  is\\ any\\ rational\\ number\\  then\\  \\hspace{15cm}\\]<\/div>\n\n\n\n<div class=\"wp-block-mathml-mathmlblock\">\\[(1\\ +\\ x)^n\\ =\\ 1\\ +\\ nx\\ +\\ \\frac{n(n-1)}{1.\\ 2}\\ x^2\\ +\\ \\frac{n(n-1)(n-2)}{1.\\ 2.\\ 3}\\ x^3\\ +\\ &#8230;&#8230;.\\]<\/div>\n\n\n\n<script async src=\"https:\/\/pagead2.googlesyndication.com\/pagead\/js\/adsbygoogle.js?client=ca-pub-9453835310745500\"\n     crossorigin=\"anonymous\"><\/script>\n<!-- Leader board 1 -->\n<ins class=\"adsbygoogle\"\n     style=\"display:inline-block;width:728px;height:90px\"\n     data-ad-client=\"ca-pub-9453835310745500\"\n     data-ad-slot=\"8769628924\"><\/ins>\n<script>\n     (adsbygoogle = window.adsbygoogle || []).push({});\n<\/script>\n","protected":false},"excerpt":{"rendered":"<p>A square matrix A is called a singular matrix<\/p>\n","protected":false},"author":187055548,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center 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