{"id":5063,"date":"2021-09-05T16:06:57","date_gmt":"2021-09-05T10:36:57","guid":{"rendered":"https:\/\/mathemerize.com\/?p=5063"},"modified":"2021-11-21T16:15:54","modified_gmt":"2021-11-21T10:45:54","slug":"properties-of-determinant-of-matrix","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/properties-of-determinant-of-matrix\/","title":{"rendered":"Properties of Determinant of Matrix Class 12"},"content":{"rendered":"

Here you will learn properties of determinant of matrix with examples.<\/p>\n

Let’s begin –<\/p>\n

Properties of Determinant of Matrix<\/h2>\n

Property 1 :<\/strong><\/h4>\n

The value of determinant remains unaltered or unchanged, if the rows & columns are inter-changed,<\/p>\n

e.g. if D = \\(\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\) = \\(\\begin{bmatrix} a_1 & a_2 & a_3 \\\\ b_1 & b_2 & b_3 \\\\ c_1 & c_2 & c_3 \\end{bmatrix}\\)<\/p>\n

Property 2 :<\/strong><\/h4>\n

If any two rows (or columns) of a determinant be interchanged, the value of determinant is changed in sign only. e.g.<\/p>\n

Let D = \\(\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\) & \\(D_1\\) = \\(\\begin{bmatrix}  a_2 & b_2 & c_2 \\\\ a_1 & b_1 & c_1  \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\). Then \\(D_1\\) = -D.<\/p>\n

Property 3 :<\/strong><\/h4>\n

If all the elements of a row (or column) are zero, then the value of the determinant is zero.<\/p>\n

Property 4 :<\/strong><\/h4>\n

If all the elements of any row (or column) are multiplied by the same number, then the determinant is multiplied by that number.<\/p>\n

e.g.  If D = \\(\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\) and \\(D_1\\) = \\(\\begin{bmatrix} ka_1 & kb_1 & kc_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\). Then \\(D_1\\) = k D.<\/p>\n

Property 5 :<\/strong><\/h4>\n

If all the elements of a row (or column) are proportional (or identical) to the elements of any other row, then the determinant vanishes, i.e. its value is zero.<\/p>\n

e.g. If D = \\(\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\) \\(\\implies\\) D = 0<\/p>\n

If \\(D_1\\)=- \\(\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ ka_1 & kb_1 & kc_1 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\). \\(\\implies\\) \\(D_1\\) = 0.<\/p>\n

Property 6 :<\/strong><\/h4>\n

If each element of any row (or column) is expressed as sum of two (or more) terms, then the determinant can be expressed as the sum of two (or more) determinants.<\/p>\n

e.g. \\(\\begin{bmatrix} a_1 + x & b_1 + y & c_1 + z \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\) = \\(\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\) + \\(\\begin{bmatrix} x & y & z \\\\ a_2 & b_2 & c_2 \\\\ a_3 & b_3 & c_3 \\end{bmatrix}\\)<\/p>\n

Property 7 :<\/strong><\/h4>\n

Row – column Operation<\/strong> : The value of a determinant remains unaltered under a column (\\(C_i\\)) operation of the form \\(C_i\\) \\(\\rightarrow\\) \\(C_i\\) + \\(\\alpha C_j\\)  (j \\(\\ne\\) i) or row (\\(R_i\\)) operation of the form \\(R_i\\) \\(\\rightarrow\\) \\(R_i\\) + \\(\\alpha R_j\\)  (j \\(\\ne\\) i). In other words, the value of determinant  is not altered by adding elements of any row ( or column) to the same multiples of the corresponding elements of any other row (or column).<\/p>\n

Property 8 :<\/strong><\/h4>\n

If the elements of a determinant D are rational integral functions of x and two rows (or columns) become identical when x = a then (x – a) is a factor of D.<\/p>\n

Note that if rows become identical when a is substituted for x, then \\((x-a)^{r-1}\\) is a factor of D.<\/p>\n\n\n

\n
Next – Minors and Cofactors of a Matrix (3\u00d73 and 2\u00d72)<\/a><\/div>\n<\/div>\n\n\n\n
\n
Previous – Determinants of Matrix 4×4<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn properties of determinant of matrix with examples. Let’s begin – Properties of Determinant of Matrix Property 1 : The value of determinant remains unaltered or unchanged, if the rows & columns are inter-changed, e.g. if D = \\(\\begin{bmatrix} a_1 & b_1 & c_1 \\\\ a_2 & b_2 & c_2 \\\\ a_3 …<\/p>\n

Properties of Determinant of Matrix Class 12<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","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":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[35],"tags":[210,211,219,220],"yoast_head":"\nProperties of Determinant of Matrix Class 12 - Mathemerize<\/title>\n<meta name=\"description\" content=\"In this post you will learn properties of determinant of matrix class 12 with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/properties-of-determinant-of-matrix\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Properties of Determinant of Matrix Class 12 - 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