{"id":19995,"date":"2024-04-15T05:46:58","date_gmt":"2024-04-15T05:46:58","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=19995"},"modified":"2024-04-15T05:46:58","modified_gmt":"2024-04-15T05:46:58","slug":"which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-statements-is-true-about-every-n-a%c2%97-n-matrix-with-only-real-eigen-values-a-if-the-trace-of-the-matrix-is-positive-and-the-determinant-of-the-matrix-is-negative-at-le\/","title":{"rendered":"Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. If the product of the trace and determinant of the matrix is positive, all its eigen values are positive"},"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_6a99176f7216c\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"true\">\r\n                    If the trace of the matrix is positive, all its eigen values are positive                <\/div>\r\n                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    If the product of the trace and determinant of the matrix is positive, all its eigen values are positive                <\/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                        \/\/ 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<!--more--><\/p>\n<p>The correct answer is $\\boxed{\\text{A}}$.<\/p>\n<p>The trace of a matrix is the sum of its eigenvalues, and the determinant of a <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> matrix is the product of its eigenvalues. If the trace of a matrix is positive and the determinant of the matrix is negative, then at least one of its eigenvalues must be negative. This is because the trace and determinant of a matrix are both invariants of the matrix, meaning that they do not change when the matrix is transformed by a similarity transformation. Therefore, if the trace and determinant of a matrix have different signs, then the matrix must have at least one eigenvalue with a negative real part.<\/p>\n<p>The other options are incorrect. Option $\\text{B}$ is incorrect because it is possible for a matrix to have a positive trace and negative eigenvalues. For example, the matrix $$A = \\begin{bmatrix} 1 &amp; 2 \\\\ 0 &amp; -1 \\end{bmatrix}$$ has a trace of 1 and a determinant of -1, but its eigenvalues are both negative. Option $\\text{C}$ is incorrect because it is possible for a matrix to have a positive determinant and negative eigenvalues. For example, the matrix $$B = \\begin{bmatrix} 1 &amp; 0 \\\\ 0 &amp; -2 \\end{bmatrix}$$ has a determinant of 2 and a negative eigenvalue. Option $\\text{D}$ is incorrect because it is possible for a matrix to have a positive product of the trace and determinant and negative eigenvalues. For example, the matrix $$C = \\begin{bmatrix} 1 &amp; 2 \\\\ 0 &amp; -3 \\end{bmatrix}$$ has a product of the trace and determinant of 3 and a negative eigenvalue.<\/p>\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>Which one of the following statements is TRUE about every n \u00c3\u0097 n matrix with only real eigen values? A. If the trace of the matrix is positive and the determinant of the matrix is negative, at least one of its eigen values is negative B. If the trace of the matrix is positive, all its eigen values are positive C. 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If the trace of the matrix is positive, all its eigen values are positive C. If the determinant of the matrix is positive, all its eigen values are positive D. 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