{"id":20065,"date":"2024-04-15T05:47:53","date_gmt":"2024-04-15T05:47:53","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20065"},"modified":"2024-04-15T05:47:53","modified_gmt":"2024-04-15T05:47:53","slug":"which-one-of-the-following-equations-is-a-correct-identity-for-arbitrary-3-a%c2%97-3-real-matrices-p-q-and-r-a-pq-r-pq-rp-b-p-q2-p2-2pq-q2-c-det-p-q-det-p-det-q-d-p","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/which-one-of-the-following-equations-is-a-correct-identity-for-arbitrary-3-a%c2%97-3-real-matrices-p-q-and-r-a-pq-r-pq-rp-b-p-q2-p2-2pq-q2-c-det-p-q-det-p-det-q-d-p\/","title":{"rendered":"Which one of the following equations is a correct identity for arbitrary 3 \u00c3\u0097 3 real matrices P, Q and R? A. P(Q + R) = PQ + RP B. (P &#8211; Q)2 = P2 &#8211; 2PQ + Q2 C. det (P + Q) = det P + det Q D. (P + Q)2 = P2 + PQ + QP + Q2"},"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_6a9a7b2763325\">\r\n                                            <div class=\"option\" data-option-key=\"option1\" data-is-correct=\"false\">\r\n                    P(Q + R) = PQ + RP                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option2\" data-is-correct=\"false\">\r\n                    (P - Q)2 = P2 - 2PQ + Q2                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option3\" data-is-correct=\"true\">\r\n                    det (P + Q) = det P + det Q                <\/div>\r\n                                            <div class=\"option\" data-option-key=\"option4\" data-is-correct=\"false\">\r\n                    (P + Q)2 = P2 + PQ + QP + Q2                <\/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 $\\boxed{\\text{(D)}}$.<\/p>\n<p>(A) is not correct because the order of multiplication of matrices does not matter. In other words, $P(Q + R) \\neq PQ + PR$.<\/p>\n<p>(B) is not correct because the square of a matrix is not equal to the sum of the matrix and its negative. In other words, $(P &#8211; Q)^2 \\neq P^2 &#8211; 2PQ + Q^2$.<\/p>\n<p>(C) is not correct because the determinant of a matrix is not equal to the sum of the determinants of its diagonal elements. In other words, $\\det (P + Q) \\neq \\det P + \\det Q$.<\/p>\n<p>(D) is correct because the square of a matrix is equal to the sum of the product of the matrix and itself, its transpose, and its negative transpose. In other words, $(P + Q)^2 = P^2 + 2PQ + QP + Q^2$.<\/p>\n<p>Here is a more detailed explanation of each option:<\/p>\n<p>(A) $P(Q + R) = PQ + RP$<\/p>\n<p>This is not always true. For example, if $P = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix}$, $Q = \\begin{bmatrix} 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\\\ 1 &amp; 0 &amp; 0 \\end{bmatrix}$, and $R = \\begin{bmatrix} 0 &amp; 0 &amp; 1 \\\\ 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\end{bmatrix}$, then $P(Q + R) = \\begin{bmatrix} 1 &amp; 1 &amp; 1 <div class=\"telegram-channel-container\">\r\n        <a href=\"https:\/\/t.me\/pscnotes2025\" target=\"_blank\" class=\"telegram-channel-button\">\r\n            <span class=\"telegram-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 496 512\">\r\n                    <path fill=\"white\" d=\"M248,8C111,8,0,119,0,256s111,248,248,248s248-111,248-248S385,8,248,8z M362,177L320,367c-3,14-10,18-20,14l-56-41l-27,26 c-3,3-5,5-10,5l4-63L323,196c5-5-1-7-8-3l-98,62l-42-13c-9-3-10-9,2-14l162-63C351,160,365,164,362,177z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Join Our Telegram Channel\r\n        <\/a>\r\n    <\/div> \\\\ 1 &amp; 1 &amp; 1 \\\\ 1 &amp; 1 &amp; 1 \\end{bmatrix}$, but $PQ + PR = \\begin{bmatrix} 1 &amp; 1 &amp; 1 \\\\ 1 &amp; 1 &amp; 1 \\\\ 0 &amp; 1 &amp; 0 \\end{bmatrix}$.<\/p>\n<p>(B) $(P &#8211; Q)^2 = P^2 &#8211; 2PQ + Q^2$<\/p>\n<p>This is not always true. For example, if $P = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; <div class=\"youtube-subscribe-container\">\r\n        <a href=\"https:\/\/www.youtube.com\/channel\/UCNHT8lW-JmLC68rjBfZhdkg?sub_confirmation=1\" target=\"_blank\" class=\"youtube-subscribe-button\">\r\n            <span class=\"youtube-icon\">\r\n                <svg xmlns=\"http:\/\/www.w3.org\/2000\/svg\" viewBox=\"0 0 576 512\">\r\n                    <path d=\"M549.7 124.1c-6.3-23.7-24.8-42.3-48.3-48.6C458.8 64 288 64 288 64S117.2 64 74.6 75.5c-23.5 6.3-42 24.9-48.3 48.6-11.4 42.9-11.4 132.3-11.4 132.3s0 89.4 11.4 132.3c6.3 23.7 24.8 41.5 48.3 47.8C117.2 448 288 448 288 448s170.8 0 213.4-11.5c23.5-6.3 42-24.2 48.3-47.8 11.4-42.9 11.4-132.3 11.4-132.3s0-89.4-11.4-132.3zm-317.5 213.5V175.2l142.7 81.2-142.7 81.2z\"\/>\r\n                <\/svg>\r\n            <\/span>\r\n            Subscribe on YouTube\r\n        <\/a>\r\n    <\/div> 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix}$, $Q = \\begin{bmatrix} 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\\\ 1 &amp; 0 &amp; 0 \\end{bmatrix}$, then $(P &#8211; Q)^2 = \\begin{bmatrix} 1 &amp; -1 &amp; 0 \\\\ -1 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix}$, but $P^2 &#8211; 2PQ + Q^2 = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\end{bmatrix}$.<\/p>\n<p>(C) $\\det (P + Q) = \\det P + \\det Q$<\/p>\n<p>This is not always true. For example, if $P = \\begin{bmatrix} 1 &amp; 0 &amp; 0 \\\\ 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; -1 \\end{bmatrix}$ and $Q = \\begin{bmatrix} 0 &amp; 1 &amp; 0 \\\\ 0 &amp; 0 &amp; 1 \\\\ 1 &amp; 0 &amp; 0 \\end{bmatrix}$, then $\\det (P + Q) = -1$, but $\\det P + \\det Q = 2$.<\/p>\n<p>(D) $(P + Q)^2 = P^2 + 2PQ + QP + Q^2$<\/p>\n<p>This is always true. To prove this, we can use the following identity:<\/p>\n<p>$$(A + B)^2 = A^2 + 2AB + B^2$$<\/p>\n<p>where $A$ and $B$ are any matrices. Substituting $P$ for $A$ and $Q$ for $B$, we get:<\/p>\n<p>$$(P + Q)^2 = P^2 + 2PQ + Q^2$$<\/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 equations is a correct identity for arbitrary 3 \u00c3\u0097 3 real matrices P, Q and R? A. P(Q + R) = PQ + RP B. (P - Q)2 = P2 - 2PQ + Q2 C. det (P + Q) = det P + det Q D. 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