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 co-factors of $a_1,b_1,c_1$ respectively, then teh value os 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

The correct option is 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 co-factors of $a_1,b_1,c_1$ respectively.

Now,

$\left|\begin{array}{ccc}{A}_1& B_1& C_1\\ A_2& B_2& C_2\\ A_3& B_3& C_3\end{array}\right|$

$=|adj\Delta |$

$={\Delta }^{3-1}$

$={\Delta }^2$.