3 and 4 .Determinants and Matrices
hard

यदि $a_i^2 + b_i^2 + c_i^2 = 1,\,\,(i = 1,2,3)$ और ${a_i}{a_j} + {b_i}{b_j} + {c_i}{c_j} = 0$ $(i \ne j,i,j = 1,2,3)$ तब ${\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}\,} \right|^2}$ का मान है

A

$0$

B

$1/2$

C

$1$

D

$2$

Solution

${\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\\{{c_1}}&{{c_2}}&{{c_3}}\end{array}\,} \right|^2} = \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|\,\left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$,

                                                                             $[\because \,|A| = |A'|]$

$ = \left| {\,\begin{array}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{array}\,} \right| = 1$.

Standard 12
Mathematics

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