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Question: If \(\left| \begin{matrix} - a^{2} & ab & ac \\ ab & - b^{2} & bc \\ ac & bc & - c^{2} \end{matrix}...

If $\left| \begin{matrix}

  • a^{2} & ab & ac \ ab & - b^{2} & bc \ ac & bc & - c^{2} \end{matrix} \right| = Ka^{2}b^{2}c^{2},thenthenK =$
A

– 4

B

2

C

4

D

8

Answer

4

Explanation

Solution

∣−a2abacab−b2bcacbc−c2∣=abc∣−abca−bcab−c∣\left| \begin{matrix} \mathbf{-}\mathbf{a}^{\mathbf{2}} & \mathbf{ab} & \mathbf{ac} \\ \mathbf{ab} & \mathbf{-}\mathbf{b}^{\mathbf{2}} & \mathbf{bc} \\ \mathbf{ac} & \mathbf{bc} & \mathbf{-}\mathbf{c}^{\mathbf{2}} \end{matrix} \right|\mathbf{= abc}\left| \begin{matrix} \mathbf{-}\mathbf{a} & \mathbf{b} & \mathbf{c} \\ \mathbf{a} & \mathbf{-}\mathbf{b} & \mathbf{c} \\ \mathbf{a} & \mathbf{b} & \mathbf{-}\mathbf{c} \end{matrix} \right|

\mathbf{-}\mathbf{1} & \mathbf{1} & \mathbf{1} \\ \mathbf{1} & \mathbf{-}\mathbf{1} & \mathbf{1} \\ \mathbf{1} & \mathbf{1} & \mathbf{-}\mathbf{1} \end{matrix} \right|\mathbf{=}\mathbf{a}^{\mathbf{2}}\mathbf{b}^{\mathbf{2}}\mathbf{c}^{\mathbf{2}}\mathbf{(}\mathbf{-}\mathbf{1)(}\mathbf{-}\mathbf{4)}$$ $= 4a^{2}b^{2}c^{2} = Ka^{2}b^{2}c^{2}$, (given) ⇒ K = 4.