{"id":20045,"date":"2024-04-15T05:47:37","date_gmt":"2024-04-15T05:47:37","guid":{"rendered":"https:\/\/exam.pscnotes.com\/mcq\/?p=20045"},"modified":"2024-04-15T05:47:37","modified_gmt":"2024-04-15T05:47:37","slug":"if-the-following-system-has-non-trivial-solution-px-qy-rz-0-qx-ry-pz-0-rx-py-qz-0-then-which-one-of-the-following-options-is-true-a-p-q-r-0-or-p-q-r-b-p-q-r-0-or","status":"publish","type":"post","link":"https:\/\/exam.pscnotes.com\/mcq\/if-the-following-system-has-non-trivial-solution-px-qy-rz-0-qx-ry-pz-0-rx-py-qz-0-then-which-one-of-the-following-options-is-true-a-p-q-r-0-or-p-q-r-b-p-q-r-0-or\/","title":{"rendered":"If the following system has non-trivial solution, px + qy + rz = 0 qx + ry + pz = 0 rx + py + qz = 0 then which one of the following options is TRUE? A. p – q + r = 0 or p = q = -r B. p + q – r = 0 or p = -q = r C. p + q + r = 0 or p = q = r D. p – q + r = 0 or p = -q = -r"},"content":{"rendered":"

[amp_mcq option1=”p – q + r = 0 or p = q = -r” option2=”p + q – r = 0 or p = -q = r” option3=”p + q + r = 0 or p = q = r” option4=”p – q + r = 0 or p = -q = -r” correct=”option3″]<\/p>\n

The correct answer is: $\\boxed{\\text{A. }p – q + r = 0 \\text{ or } p = q = -r}$.<\/p>\n

To solve this, we can use the following theorem:<\/p>\n

\n

If a system of equations has a non-trivial solution, then the determinant of the coefficient matrix is equal to zero.<\/p>\n<\/blockquote>\n

In this case, the coefficient matrix is:<\/p>\n

$$\\begin{bmatrix}
\np & q & r \\
\nq & r & p \\
\nr & p & q
\n\\end{bmatrix}$$<\/p>\n

The determinant of this matrix is:<\/p>\n

$$p^2 + q^2 + r^2 – 2pq – 2pr – 2qr$$<\/p>\n

If this determinant is equal to zero, then the system of equations has a non-trivial solution.<\/p>\n

We can factor the determinant as follows:<\/p>\n

$$p^2 + q^2 + r^2 – 2pq – 2pr – 2qr = (p – q + r)(p + q – r)(p + q + r)$$<\/p>\n

Therefore, the system of equations has a non-trivial solution if and only if one of the following three equations is true:<\/p>\n

$$p – q + r = 0$$
\n$$p + q – r = 0$$
\n$$p + q + r = 0$$<\/p>\n

The first equation is equivalent to $p = q = -r$. The second equation is equivalent to $p = -q = r$. The third equation is equivalent to $p = q = r$.<\/p>\n

Therefore, the correct answer is $\\boxed{\\text{A. }p – q + r = 0 \\text{ or } p = q = -r}$.<\/p>\n","protected":false},"excerpt":{"rendered":"

[amp_mcq option1=”p – q + r = 0 or p = q = -r” option2=”p + q – r = 0 or p = -q = r” option3=”p + q + r = 0 or p = q = r” option4=”p – q + r = 0 or p = -q = -r” correct=”option3″]<\/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":[],"class_list":["post-20045","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","no-featured-image-padding"],"yoast_head":"\nIf the following system has non-trivial solution, px + qy + rz = 0 qx + ry + pz = 0 rx + py + qz = 0 then which one of the following options is TRUE? A. p - q + r = 0 or p = q = -r B. p + q - r = 0 or p = -q = r C. p + q + r = 0 or p = q = r D. p - q + r = 0 or p = -q = -r<\/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\/if-the-following-system-has-non-trivial-solution-px-qy-rz-0-qx-ry-pz-0-rx-py-qz-0-then-which-one-of-the-following-options-is-true-a-p-q-r-0-or-p-q-r-b-p-q-r-0-or\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"If the following system has non-trivial solution, px + qy + rz = 0 qx + ry + pz = 0 rx + py + qz = 0 then which one of the following options is TRUE? 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A. p - q + r = 0 or p = q = -r B. p + q - r = 0 or p = -q = r C. p + q + r = 0 or p = q = r D. p - q + r = 0 or p = -q = -r","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/exam.pscnotes.com\/mcq\/if-the-following-system-has-non-trivial-solution-px-qy-rz-0-qx-ry-pz-0-rx-py-qz-0-then-which-one-of-the-following-options-is-true-a-p-q-r-0-or-p-q-r-b-p-q-r-0-or\/","og_locale":"en_US","og_type":"article","og_title":"If the following system has non-trivial solution, px + qy + rz = 0 qx + ry + pz = 0 rx + py + qz = 0 then which one of the following options is TRUE? 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