{"id":20038,"date":"2024-04-15T05:47:32","date_gmt":"2024-04-15T05:47:32","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20038"},"modified":"2024-04-15T05:47:32","modified_gmt":"2024-04-15T05:47:32","slug":"the-eigen-values-of-a-skew-symmetric-matrix-are-a-always-zero-b-always-pure-imaginary-c-either-zero-or-pure-imaginary-d-always-real","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/the-eigen-values-of-a-skew-symmetric-matrix-are-a-always-zero-b-always-pure-imaginary-c-either-zero-or-pure-imaginary-d-always-real\/","title":{"rendered":"The eigen values of a skew-symmetric matrix are A. always zero B. always pure imaginary C. either zero or pure imaginary D. always real"},"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_6a9b2105dfced\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    always zero                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    always pure imaginary                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    either zero or pure imaginary                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    always real                <\/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    <!--more--><\/p>\n<p>The correct answer is: C. either zero or pure imaginary.<\/p>\n<p>A skew-symmetric matrix is a square matrix $A$ such that $A^T = -A$. The eigenvalues of a skew-symmetric matrix <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> are always either zero or pure imaginary. This can be shown by using the fact that the characteristic polynomial of a skew-symmetric matrix is of the form $p(x) = x^2 &#8211; tr(A)x + \\det(A)$. Since $tr(A) = 0$ for a skew-symmetric matrix, the characteristic polynomial has two distinct roots, which must be either both real or both complex. If the roots are real, then they must be both zero. If the roots are complex, then they must be a complex conjugate pair.<\/p>\n<p>Here is a brief explanation of each option:<\/p>\n<ul>\n<li>Option A: The eigenvalues of a skew-symmetric matrix are always zero. This is not always true, as shown above.<\/li>\n<li>Option B: The eigenvalues of a skew-symmetric matrix are always pure imaginary. This is also not always true, as shown above.<\/li>\n<li>Option C: The eigenvalues of a skew-symmetric matrix are either zero or <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> pure imaginary. This is the correct answer, as shown above.<\/li>\n<li>Option D: The eigenvalues of a skew-symmetric matrix are always real. This is not always true, as shown above.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>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>The eigen values of a skew-symmetric matrix are A. always zero B. always pure imaginary C. either zero or pure imaginary D. always real<\/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\/the-eigen-values-of-a-skew-symmetric-matrix-are-a-always-zero-b-always-pure-imaginary-c-either-zero-or-pure-imaginary-d-always-real\/\" \/>\n<meta property=\"og:locale\" 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