a
(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 $.