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NDA 2012
Exam
NDA
Year
2012
Subject
Mathematics
Chapter
Determinants

NDA 2012 · Mathematics · Determinants

What is | array ccc -a^2 & a b & a c a b & -b^2 & b c a c & b c &...

What is \left|−a2abacab−b2bcacbc−c2\begin{array}{ccc}-a^2 & a b & a c \\ a b & -b^2 & b c \\ a c & b c & -c^2\end{array}\right| equal to?

Options

  1. A4abc4 a b c
  2. B4a2bc4 a^2 b c
  3. C4a2b2c24 a^2 b^2 c^2
  4. D−4a2b2c2-4 a^2 b^2 c^2

Correct answer

C. 4a2b2c24 a^2 b^2 c^2

Step-by-step solution

∣−a2abacab−b2bcacbc−c2∣=abc∣−abca−bcab−c∣=a2b2c2∣−1111−1111−1∣=a2b2c2∣−111002020∣=a2b2c2[(−1)(−4)]=4a2b2c2\begin{aligned} & \left|\begin{array}{ccc}-a^2 & a b & a c \\ a b & -b^2 & b c \\ a c & b c & -c^2\end{array}\right| \\ & =a b c\left|\begin{array}{ccc}-a & b & c \\ a & -b & c \\ a & b & -c\end{array}\right| \\ & =a^2 b^2 c^2\left|\begin{array}{ccc}-1 & 1 & 1 \\ 1 & -1 & 1 \\ 1 & 1 & -1\end{array}\right| \\ & =a^2 b^2 c^2\left|\begin{array}{ccc}-1 & 1 & 1 \\ 0 & 0 & 2 \\ 0 & 2 & 0\end{array}\right| \\ & =a^2 b^2 c^2[(-1)(-4)] \\ & =4 a^2 b^2 c^2\end{aligned}
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