A) \[\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\]
B) \[\left| \begin{matrix} {{d}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{d}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{d}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\]
C) \[\left| \begin{matrix} {{d}_{1}} & {{a}_{1}} & {{c}_{1}} \\ {{d}_{2}} & {{a}_{2}} & {{c}_{2}} \\ {{d}_{3}} & {{a}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\]
D) \[\left| \begin{matrix} {{d}_{1}} & {{a}_{1}} & {{b}_{1}} \\ {{d}_{2}} & {{a}_{2}} & {{b}_{2}} \\ {{d}_{3}} & {{a}_{3}} & {{b}_{3}} \\ \end{matrix} \right|=0\]
Correct Answer: B
Solution :
[b] The given system of equations is \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}z={{d}_{1}}\] \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}z={{d}_{2}}\] and \[{{a}_{3}}x+{{b}_{3}}y+{{c}_{3}}z={{d}_{3}}\] Let \[\Delta =\left| \begin{matrix} {{a}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{a}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{a}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|\] This system has a unique solution \[{{x}_{0}},{{y}_{0}},{{z}_{0}}\] If \[\Delta \ne 0\] and \[{{x}_{0}}=\frac{\Delta x}{\Delta }=0\Rightarrow \Delta x=0\] \[\Rightarrow \left| \begin{matrix} {{d}_{1}} & {{b}_{1}} & {{c}_{1}} \\ {{d}_{2}} & {{b}_{2}} & {{c}_{2}} \\ {{d}_{3}} & {{b}_{3}} & {{c}_{3}} \\ \end{matrix} \right|=0\]
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