If $Λ = |\begin{array}{lll} a_{1} b_{1} c_{1} \\ a_{2} b_{2} c_{2} \\ a_{3} b_{3} c_{3} \end{array}| and A_{1}, B_{1}, C_{1}$ denote the cofactors of $a_{1}, b_{1}, c_{1}$ respectively, then the value of the determinant $|\begin{array}{lll} A_{1} B_{1} C_{1} \\ A_{2} B_{2} C_{2} \\ A_{3} B_{3} C_{3} \end{array}|$ is

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$∆$

$∆^{2}$

$∆^{3}$

answer is B.

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Detailed Solution

We know that $Δ . Δ^{'} = |\begin{array}{lll} a_{1} b_{1} c_{1} \\ a_{2} b_{2} c_{2} \\ a_{3} b_{3} c_{3} \end{array}| \cdot |\begin{array}{lll} A_{1} B_{1} C_{1} \\ A_{2} B_{2} C_{2} \\ A_{3} B_{3} C_{3} \end{array}|$

$\begin{array}{l} = |\begin{array}{ccc} Σ a_{1} A_{1} & 0 & 0 \\ 0 & Σ a_{2} A_{2} & 0 \\ 0 & 0 & Σ a_{3} A_{3} \end{array}| = |\begin{array}{ccc} Δ & 0 & 0 \\ 0 & Δ & 0 \\ 0 & 0 & Δ \end{array}| = Δ^{3} \\ \Rightarrow Δ^{'} = Δ^{2} \end{array}$