Question

If $\Delta =\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$ and $A_1,B_1,C_1$ denote the cofactors of $a_1,b_1,c_1$ respectively, then the value of the determinant $\left|\begin{array}{ccc}{A}_1& B_1& C_1\\ A_2& B_2& C_2\\ A_3& B_3& C_3\end{array}\right|$ is

• A. $\Delta$
• B. ${\Delta }^2$
• C. ${\Delta }^3$
• D. 0

Answer 1

Answer: (B)

${\Delta }^2$

Given, $\Delta =\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|$ and $A_1,B_1,C_1$ denote the cofactors of $a_1,b_1,c_1$ respectively.
let $A=\left[\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right]$
Cofactor matrix $C=\left[\begin{array}{ccc}{A}_1& B_1& C_1\\ A_2& B_2& C_2\\ A_3& B_3& C_3\end{array}\right]$
$adjA=C^T=\left[\begin{array}{ccc}{A}_1& A_2& A_3\\ B_1& B_2& B_3\\ C_1& C_2& C_3\end{array}\right]$
But $A\left(adjA\right)=|A|I$
$\Rightarrow |A||adjA|=|A{|}^3$
$\Rightarrow |adjA|=|A{|}^2$
$\therefore |C|=|A{|}^2$ Since,$|A|=|A^T|$
$\therefore \left|\begin{array}{ccc}{A}_1& B_1& C_1\\ A_2& B_2& C_2\\ A_3& B_3& C_3\end{array}\right|={\Delta }^2$
Hence, option B.