{"id":20114,"date":"2024-04-15T05:48:35","date_gmt":"2024-04-15T05:48:35","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20114"},"modified":"2024-04-15T05:48:35","modified_gmt":"2024-04-15T05:48:35","slug":"if-any-two-columns-of-a-determinant-textp-left-beginarray20c-478-315-962-endarray-right-are-interchanged-which-one-of-the-following-statements-reg","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/if-any-two-columns-of-a-determinant-textp-left-beginarray20c-478-315-962-endarray-right-are-interchanged-which-one-of-the-following-statements-reg\/","title":{"rendered":"If any two columns of a determinant \\[{\\text{P}} = \\left| {\\begin{array}{*{20}{c}} 4&#038;7&#038;8 \\\\ 3&#038;1&#038;5 \\\\ 9&#038;6&#038;2 \\end{array}} \\right|\\] are interchanged, which one of the following statements regarding the value of the determinant is CORRECT? A. Absolute value remains unchanged but sign will change B. Both absolute value and sign will change C. Absolute value will change but sign will not change D. Both absolute value and sign will remain unchanged"},"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_6a97718f562fc\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"true\">\r\n                    Absolute value remains unchanged but sign will change                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    Both absolute value and sign will change                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"false\">\r\n                    Absolute value will change but sign will not change                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    Both absolute value and sign will remain unchanged                <\/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 A. Absolute value remains unchanged but sign will change.<\/p>\n<p>The determinant of a 3&#215;3 matrix can be computed using the following formula:<\/p>\n<p>$$\\det(A) = \\sum_{i=1}^3 (-1)^{i+1} a_{1i} \\det(A_{i,i})$$<\/p>\n<p>where $A_{i,i}$ is the determinant of the matrix that is obtained from $A$ by deleting row $i$ and column $i$.<\/p>\n<p>If any two columns of $A$ are interchanged, then the sign of the determinant will change. This is because the determinant of $A_{i,i}$ will be equal to the determinant of $A_{i&#8217;,i&#8217;}$, where $i&#8217;$ is the index of the column that was interchanged with column $i$. However, the order of the terms in the sum in the formula for the determinant will be different, so the sign of the determinant will change.<\/p>\n<p>The absolute value of the determinant will not change, because the determinant of a matrix is always a real number.<\/p>\n<p>Here is an example to illustrate this. Let $A$ be the following matrix:<\/p>\n<p>$$A = \\left| {\\begin{array}{*{20}{c}} 1&amp;2&amp;3 \\ 4&amp;5&amp;6 \\ 7&amp;8&amp;9 \\end{array}} \\right|$$<\/p>\n<p>If we interchange columns 1 and 2, then the new matrix is:<\/p>\n<p>$$B = \\left| {\\begin{array}{*{20}{c}} 4&amp;2&amp;3 \\ 1&amp;5&amp;6 \\ 7&amp;8&amp;9 \\end{array}} \\right|$$<\/p>\n<p>The determinant of $A$ is equal to $-30$, and the determinant of $B$ <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   <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>          <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> is also equal to $-30$. However, the sign of the determinant of $B$ is negative, because the order of the terms in the sum in the formula for the determinant is different.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>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>If any two columns of a determinant \\[{\\text{P}} = \\left| {\\begin{array}{*{20}{c}} 4&amp;7&amp;8 \\\\ 3&amp;1&amp;5 \\\\ 9&amp;6&amp;2 \\end{array}} \\right|\\] are interchanged, which one of the following statements regarding the value of the determinant is CORRECT? A. Absolute value remains unchanged but sign will change B. Both absolute value and sign will change C. 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