Question

If $A_{1},B_{1},C_{1}$.... are respectively the co-factors of the elements $a_{1},b_{1},c_{1}$,...... of the determinant.$\Delta = \left| \begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix} \right|$, then $\left| \begin{matrix} B_{2} & C_{2} \\ B_{3} & C_{3} \end{matrix} \right| =$

• A. $a_{1}\Delta$
• B. $a_{1}a_{3}\Delta$
• C. $(a_{1} + b_{1})\Delta$
• D. None of these

Answer 1

Answer: (A)

$a_{1}\Delta$

Answer 2

$B_{2} = \left| \begin{matrix} a_{1} & c_{1} \\ a_{3} & c_{3} \end{matrix} \right| = a_{1}c_{3} - c_{1}a_{3}$

a_{1} & b_{1} \\ a_{3} & b_{3} \end{matrix} \right| = - (a_{1}b_{3} - a_{3}b_{1})$$ $$B_{3} = - \left| \begin{matrix} a_{1} & c_{1} \\ a_{2} & c_{2} \end{matrix} \right| = - (a_{1}c_{2} - a_{2}c_{1})$$ $$C_{3} = \left| \begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix} \right| = a_{1}b_{2} - a_{2}b_{1}$$ $$\left| \begin{matrix} B_{2} & C_{2} \\ B_{3} & C_{3} \end{matrix} \right| = \left| \begin{matrix} 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{matrix} \right|$$ $$= \left| \begin{matrix} a_{1}c_{3} & - a_{1}b_{3} \\ - a_{1}c_{2} & a_{1}b_{2} \end{matrix} \right| + \left| \begin{matrix} a_{1}c_{3} & a_{3}b_{1} \\ - a_{1}c_{2} & - a_{2}b_{1} \end{matrix} \right|$$ $+ \left| \begin{matrix} - a_{3}c_{1} & - a_{1}b_{3} \\ a_{2}c_{1} & a_{1}b_{2} \end{matrix} \right| + \left| \begin{matrix} - a_{3}c_{1} & a_{3}b_{1} \\ a_{2}c_{1} & - a_{2}b_{1} \end{matrix} \right|$ $$= a_{1}^{2}(b_{2}c_{3} - b_{3}c_{2}) + a_{1}b_{1}( - c_{3}a_{2} + a_{3}c_{2})$$ $+ a_{1}c_{1}( - a_{3}b_{2} + a_{2}b_{3}) + c_{1}b_{1}(a_{3}a_{2} - a_{2}a_{3}) = a_{1}\Delta$.