{"id":19132,"date":"2023-01-02T15:27:57","date_gmt":"2023-01-02T12:27:57","guid":{"rendered":"https:\/\/starlanguageblog.com\/?p=12915"},"modified":"2023-01-02T15:27:57","modified_gmt":"2023-01-02T12:27:57","slug":"does-deta-b-deta-detb-hold-2","status":"publish","type":"post","link":"https:\/\/www.starlanguageblog.com\/does-deta-b-deta-detb-hold-2\/","title":{"rendered":"Does $\\det(A + B) = \\det(A) + \\det(B)$ hold?"},"content":{"rendered":"<h1 class=\"LC20lb MBeuO DKV0Md\">Does $\\det(A + B) = \\det(A) + \\det(B)$ hold?<\/h1>\n<p>No, in general $\\det(A + B) \u2260 \\det(A) + \\det(B)$.<\/p>\n<p>The determinant of a matrix is a scalar value that can be calculated for any square matrix. It is denoted as det(A) or |A|. The determinant of a matrix is equal to the sum of the products of its elements along its main diagonal, multiplied by the alternating signs +1 and -1.<\/p>\n<p>For example, the determinant of the 2 x 2 matrix:<\/p>\n<p>[a, b] [c, d]<\/p>\n<p>is calculated as follows:<\/p>\n<p>|A| = a<em>d &#8211; b<\/em>c<\/p>\n<p>The determinant of a matrix has some important properties, including the following:<\/p>\n<ul>\n<li>The determinant of a matrix is equal to the product of its eigenvalues.<\/li>\n<li>The determinant of a matrix is invariant under invertible linear transformations.<\/li>\n<li>The determinant of a matrix is used to calculate the inverse of a matrix.<\/li>\n<\/ul>\n<p>However, the determinant does not follow the addition <a href=\"https:\/\/starlanguageblog.com\/html-code-for-new-line-with-examples\/\">property<\/a> that you mentioned. In other words, $\\det(A + B) \u2260 \\det(A) + \\det(B)$ in general.<\/p>\n<h2>Here are some examples to illustrate the fact that $\\det(A + B) \u2260 \\det(A) + \\det(B)$:<\/h2>\n<h3>Example 1:<\/h3>\n<p>Let&#8217;s consider the following 2 x 2 matrices:<\/p>\n<p>A = [1, 2] [3, 4]<\/p>\n<p>B = [5, 6] [7, 8]<\/p>\n<p>Then,<\/p>\n<p>$\\det(A) = 1<em>4 &#8211; 2<\/em>3 = -2$<\/p>\n<p>$\\det(B) = 5<em>8 &#8211; 6<\/em>7 = -2$<\/p>\n<p>$\\det(A + B) = (1+5)<em>(4+8) &#8211; (2+6)<\/em>(3+7) = -14$<\/p>\n<p>Therefore, $\\det(A + B) \u2260 \\det(A) + \\det(B)$.<\/p>\n<h3>Example 2:<\/h3>\n<p>Let&#8217;s consider the following 3 x 3 matrices:<\/p>\n<p>A = [1, 2, 3] [4, 5, 6] [7, 8, 9]<\/p>\n<p>B = [10, 11, 12] [13, 14, 15] [16, 17, 18]<\/p>\n<p>Then,<\/p>\n<p>$\\det(A) = 1*(5<em>9 &#8211; 6<\/em>8) &#8211; 2*(4<em>9 &#8211; 6<\/em>7) + 3*(4<em>8 &#8211; 5<\/em>7) = 0$<\/p>\n<p>$\\det(B) = 10*(14<em>18 &#8211; 15<\/em>17) &#8211; 11*(13<em>18 &#8211; 15<\/em>16) + 12*(13<em>17 &#8211; 14<\/em>16) = 0$<\/p>\n<p>$\\det(A + B) = (1+10)<em>((5+14)<\/em>(9+18) &#8211; (6+15)<em>(8+17)) &#8211; (2+11)<\/em>((4+14)<em>(9+18) &#8211; (6+15)<\/em>(7+16)) + (3+12)<em>((4+14)<\/em>(8+17) &#8211; (5+15)*(7+16)) = -28$<\/p>\n<p>Therefore, $\\det(A + B) \u2260 \\det(A) + \\det(B)$.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Does $\\det(A + B) = \\det(A) + \\det(B)$ hold? No, in general $\\det(A + B) \u2260 \\det(A) + \\det(B)$. The determinant of a matrix is a scalar value that can be calculated for any square matrix. It is denoted as det(A) or |A|. The determinant of a matrix is equal to the sum of the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":12916,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[2078],"tags":[2086],"class_list":["post-19132","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-maths","tag-does-deta-b-deta-detb-hold"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v23.9 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Does $\\det(A + B) = \\det(A) + \\det(B)$ hold? - Star Language Blog<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/www.starlanguageblog.com\/does-deta-b-deta-detb-hold-2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Does $\\det(A + B) = \\det(A) + \\det(B)$ hold? - Star Language Blog\" \/>\n<meta property=\"og:description\" content=\"Does $det(A + B) = det(A) + det(B)$ hold? No, in general $det(A + B) \u2260 det(A) + det(B)$. The determinant of a matrix is a scalar value that can be calculated for any square matrix. It is denoted as det(A) or |A|. 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