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નિશ્રાયક $\Delta = \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}&{{c_1}}\\{{a_2}}&{{b_2}}&{{c_2}}\\{{a_3}}&{{b_3}}&{{c_3}}\end{array}\,} \right|$ માં જો ${A_1},{B_1},{C_1}$....એ અનુક્રમે ${a_1},{b_1},{c_1}$,......ના સહઅવયવ દર્શાવે છે તો $\left| {\begin{array}{*{20}{c}}{{B_2}}&{{C_2}}\\{{B_3}}&{{C_3}}\end{array}} \right| = . . . .$
${a_1}\Delta $
${a_1}{a_3}\Delta $
$({a_1} + {b_1})\Delta $
એકપણ નહી.
Solution
(a) ${B_2} = \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{c_1}}\\{{a_3}}&{{c_3}}\end{array}} \right| = {a_1}{c_3} – {c_1}{a_3}$
${C_2} = – \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}\\{{a_3}}&{{b_3}}\end{array}\,} \right| = – ({a_1}{b_3} – {a_3}{b_1})$
${B_3} = – \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{c_1}}\\{{a_2}}&{{c_2}}\end{array}\,} \right| = – ({a_1}{c_2} – {a_2}{c_1})$
${C_3} = \left| {\,\begin{array}{*{20}{c}}{{a_1}}&{{b_1}}\\{{a_2}}&{{b_2}}\end{array}\,} \right| = {a_1}{b_2} – {a_2}{b_1}$
$\left| {\,\begin{array}{*{20}{c}}{{B_2}}&{{C_2}}\\{{B_3}}&{{C_3}}\end{array}\,} \right| = \left| {\begin{array}{*{20}{c}}{{a_1}{c_3} – {a_3}{c_1}}&{ – ({a_1}{b_3} – {a_3}{b_1})}\\{ – ({a_1}{c_2} – {a_2}{c_1})}&{{a_1}{b_2} – {a_2}{b_1}}\end{array}\,} \right|$
$ = \left| {\,\begin{array}{*{20}{c}}{{a_1}{c_3}}&{ – {a_1}{b_3}}\\{ – {a_1}{c_2}}&{{a_1}{b_2}}\end{array}\,} \right| + \left| {\,\begin{array}{*{20}{c}}{{a_1}{c_3}}&{{a_3}{b_1}}\\{ – {a_1}{c_2}}&{ – {a_2}{b_1}}\end{array}\,} \right|$
$ + \left| {\,\begin{array}{*{20}{c}}{ – {a_3}{c_1}}&{ – {a_1}{b_3}}\\{\,\,\,{a_2}{c_1}}&{{a_1}{b_2}}\end{array}\,} \right| + \left| {\,\begin{array}{*{20}{c}}{ – {a_3}{c_1}}&{{a_3}{b_1}}\\{{a_2}{c_1}}&{ – {a_2}{b_1}}\end{array}\,} \right|$
$ = a_1^2({b_2}{c_3} – {b_3}{c_2}) + {a_1}{b_1}( – {c_3}{a_2} + {a_3}{c_2})$
$ + {a_1}{c_1}( – {a_3}{b_2} + {a_2}{b_3}) + {c_1}{b_1}({a_3}{a_2} – {a_2}{a_3})$
$ = {a_1}\Delta $.
