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3 and 4 . determinant and metrices — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths3 and 4 . determinant and metrices1 Mark
MCQ
If $\Delta = \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ and ${A_1},{B_1},{C_1}$denote the co-factors of ${a_1},{b_1},{c_1}$ respectively, then the value of the determinant $\left| {\begin{array}{*{20}{c}}{{A_1}}&{{B_1}}&{{C_1}}\\{{A_2}}&{{B_2}}&{{C_2}}\\{{A_3}}&{{B_3}}&{{C_3}}\end{array}} \right|$ is
  • A
    $\Delta $
  • ✓
    ${\Delta ^2}$
  • C
    ${\Delta ^3}$
  • D
    $0$
✓

Answer

Correct option: B.
${\Delta ^2}$
b
(b) We know that $\Delta \,\Delta ' = \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|.\left| {\,\begin{array}{*{20}{c}}{{A_1}}&{{B_1}}&{{C_1}}\\{{A_2}}&{{B_2}}&{{C_2}}\\{{A_3}}&{{B_3}}&{{C_3}}\end{array}\,} \right|$

$ = \left| {\,\begin{array}{*{20}{c}}{\Sigma {a_1}{A_1}}&0&0\\0&{\Sigma {a_2}{A_2}}&0\\0&0&{\Sigma {a_3}{A_3}}\end{array}\,} \right| = \left| {\,\begin{array}{*{20}{c}}\Delta &0&0\\0&\Delta &0\\0&0&\Delta \end{array}\,} \right| = {\Delta ^3}$

==> $\Delta ' = {\Delta ^2}$.

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