{"id":20060,"date":"2024-04-15T05:47:49","date_gmt":"2024-04-15T05:47:49","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20060"},"modified":"2024-04-15T05:47:49","modified_gmt":"2024-04-15T05:47:49","slug":"a-real-n-a%c2%97-n-matrix-a-aij-is-defined-as-follows-aij-i-if-i-j-otherwise-0-the-summation-of-all-n-eigen-values-of-a-is-a-fractextnleft-textn-1-right2","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/a-real-n-a%c2%97-n-matrix-a-aij-is-defined-as-follows-aij-i-if-i-j-otherwise-0-the-summation-of-all-n-eigen-values-of-a-is-a-fractextnleft-textn-1-right2\/","title":{"rendered":"A real n \u00c3\u0097 n matrix A = {aij} is defined as follows: aij = i, if i = j, otherwise 0 The summation of all n eigen values of A is A. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)}}{2}\\] B. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} &#8211; 1} \\right)}}{2}\\] C. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)\\left( {2{\\text{n}} + 1} \\right)}}{6}\\] D. \\[{{\\text{n}}^2}\\]"},"content":{"rendered":"<p>\r\n    <!-- Check if it's an AMP page -->\r\n            <!-- Non-AMP version -->\r\n        <div class=\"mcq-container\" data-quiz-id=\"quizState_6a9ab9754c7f5\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    &#8221;[\frac{{{\text{n}}left(                <\/div>\r\n                                                                            \r\n            <!-- Feedback messages for non-AMP -->\r\n            <div class=\"feedback\" data-feedback=\"wrong\">Answer is Right!<\/div>\r\n            <div class=\"feedback\" data-feedback=\"right\">Answer is Wrong!<\/div>\r\n        <\/div>\r\n\r\n        <script>\r\n        document.addEventListener('DOMContentLoaded', function () {\r\n            var containers = document.querySelectorAll('.mcq-container');\r\n\r\n            containers.forEach(function(container) {\r\n                var options = container.querySelectorAll('.option');\r\n                var feedbackSelect = container.querySelector('[data-feedback=\"select\"]');\r\n                var feedbackWrong = container.querySelector('[data-feedback=\"wrong\"]');\r\n                var feedbackRight = container.querySelector('[data-feedback=\"right\"]');\r\n\r\n                options.forEach(function(option) {\r\n                    option.addEventListener('click', function() {\r\n                        var selectedOption = option.getAttribute('data-option-key');\r\n                        var isCorrect = option.getAttribute('data-is-correct') === 'true';\r\n\r\n                        \/\/ Remove previous selections\r\n                        options.forEach(function(opt) {\r\n                            opt.classList.remove('correct', 'incorrect');\r\n                        });\r\n\r\n                        \/\/ Add the correct\/incorrect class\r\n                        if (isCorrect) {\r\n                            option.classList.add('correct');\r\n                            feedbackRight.hidden = false;\r\n                            feedbackWrong.hidden = true;\r\n                        } else {\r\n                            option.classList.add('incorrect');\r\n                            feedbackRight.hidden = true;\r\n                            feedbackWrong.hidden = false;\r\n                        }\r\n\r\n                        \/\/ Hide select feedback\r\n                        feedbackSelect.hidden = true;\r\n                    });\r\n                });\r\n            });\r\n        });\r\n        <\/script>\r\n    \r\n    &#8221; option2=&#8221;\\[\\frac{{{\\text{n}}\\left( {{\\text{n}} &#8211; 1} \\right)}}{2}\\]&#8221; option3=&#8221;\\[\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)\\left( {2{\\text{n}} + 1} \\right)}}{6}\\]&#8221; option4=&#8221;\\[{{\\text{n}}^2}\\]&#8221; correct=&#8221;option4&#8243;]<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)}}{2}}$.<\/p>\n<p>A real n \u00c3\u0097 n matrix A = {aij} is defined as follows: aij = i, if i = j, otherwise 0. This means that the matrix is a diagonal matrix with all the diagonal elements equal to 1.<\/p>\n<p>The sum of all the eigenvalues <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> of a diagonal matrix is equal to the sum of all the diagonal elements. In this case, the sum of all the diagonal elements is n, so the sum of all the eigenvalues is $\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)}}{2}$.<\/p>\n<p>Option A is incorrect because it is the sum of the first n natural numbers. Option B is incorrect because it is the sum of the first n even natural numbers. <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> Option C is incorrect because it is the sum of the first n odd natural numbers. Option D is incorrect because it is the square of n.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>&#8221; option2=&#8221;\\[\\frac{{{\\text{n}}\\left( {{\\text{n}} &#8211; 1} \\right)}}{2}\\]&#8221; option3=&#8221;\\[\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)\\left( {2{\\text{n}} + 1} \\right)}}{6}\\]&#8221; option4=&#8221;\\[{{\\text{n}}^2}\\]&#8221; correct=&#8221;option4&#8243;] Join Our Telegram Channel Subscribe on YouTube<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[489],"tags":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO Premium plugin v22.2 (Yoast SEO v23.3) - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>A real n \u00c3\u0097 n matrix A = {aij} is defined as follows: aij = i, if i = j, otherwise 0 The summation of all n eigen values of A is A. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)}}{2}\\] B. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} - 1} \\right)}}{2}\\] C. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)\\left( {2{\\text{n}} + 1} \\right)}}{6}\\] D. \\[{{\\text{n}}^2}\\]<\/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:\/\/exam.pscnotes.com\/mcq\/a-real-n-a\u0097-n-matrix-a-aij-is-defined-as-follows-aij-i-if-i-j-otherwise-0-the-summation-of-all-n-eigen-values-of-a-is-a-fractextnleft-textn-1-right2\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A real n \u00c3\u0097 n matrix A = {aij} is defined as follows: aij = i, if i = j, otherwise 0 The summation of all n eigen values of A is A. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)}}{2}\\] B. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} - 1} \\right)}}{2}\\] C. \\[\\frac{{{\\text{n}}\\left( {{\\text{n}} + 1} \\right)\\left( {2{\\text{n}} + 1} \\right)}}{6}\\] D. \\[{{\\text{n}}^2}\\]\" \/>\n<meta property=\"og:description\" content=\"&#8221; 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