InterviewSolution
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InterviewSolution
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What is the relation between the two determinants f(x) = \(\begin{vmatrix}–a^2 & ab & ac \\ab & -b^2 & bc \\ac & bc & -c^2 \end {vmatrix}\) and g(x) = \(\begin{vmatrix}0 & c & b \\c & 0 & a \\b & a & 0 \end {vmatrix}\)?(a) f(x) = g(x)(b) f(x) = (g(x))^2(c) g(x) = 2f(x)(d) g(x) = (f(x))^2The question was asked in final exam.Question is from Application of Determinants in portion Determinants of Mathematics – Class 12 |
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Answer» CORRECT answer is (b) F(x) = (g(x))^2 Easiest explanation: Let, D = \(\begin{vmatrix}0 & c & b \\c & 0 & a \\b & a & 0 \END {vmatrix}\) Expanding D by the 1^st row we GET, D = – c\(\begin{vmatrix}c & a \\b & 0 \end {vmatrix}\) + b\(\begin{vmatrix}c & 0 \\b & a \end {vmatrix}\) = – c(0 – ab) + b(ac – 0) = 2abc Now, we have adjoint of D = D’ = \(\begin{vmatrix} \begin{vmatrix}0 & a\\ a & 0\\ \end{vmatrix} & – \begin{vmatrix}c & a\\ b & 0\\ \end{vmatrix} & \begin{vmatrix}c & 0\\ b & a\\ \end{vmatrix}\\ – \begin{vmatrix}c & b\\ a & 0\\ \end{vmatrix} & \begin{vmatrix}0 & b\\ b & 0\\ \end{vmatrix} & – \begin{vmatrix}0 & c\\ b & 0\\ \end{vmatrix} \\ \begin{vmatrix}c & b\\ 0 & a\\ \end{vmatrix} & – \begin{vmatrix}0 & b\\ c & a\\ \end{vmatrix} & \begin{vmatrix}0 & c\\ c & 0\\ \end{vmatrix} \\ \end{vmatrix}\) Or, D’ = \(\begin{vmatrix}–a^2 & ab & ac \\ab & -b^2 & bc \\ac & bc & -c^2 \end {vmatrix}\) Or, D’ = D^2 Or, D’ = D^2 = (2abc)^2 |
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