Question

Let ${\Delta }_1=\left|\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right|,$ ${\Delta }_1\ne 0$
${\Delta }_2=\left|\begin{array}{ccc}{b}_{11}& b_{12}& b_{13}\\ b_{21}& b_{22}& b_{23}\\ b_{31}& b_{32}& b_{33}\end{array}\right|$ where $b_{ij}$ is cofactor of $a_{ij}$ $\forall i,$
$j=1,2,3$
and ${\Delta }_3=\left|\begin{array}{ccc}{c}_{11}& c_{12}& c_{13}\\ c_{21}& c_{22}& c_{23}\\ c_{31}& c_{32}& c_{33}\end{array}\right|$ where $c_{ij}$ is cofactor of $b_{ij}$ $\forall i,$
$j=1,2,3,$ then which one of the following is always correct.

• A. ${\Delta }_1,{\Delta }_2,{\Delta }_3$ are in $A.P.$
• B. ${\Delta }_1,{\Delta }_2,{\Delta }_3$ are in $G.P.$
• C. ${\Delta }_1^2=\frac{{\Delta }_3}{{\Delta }_2}$
• D. ${\Delta }_1=\frac{{\Delta }_2}{\sqrt{{\Delta }_3}}$

Answer 1

The correct option is C ${\Delta }_1^2=\frac{{\Delta }_3}{{\Delta }_2}$

Given ${\Delta }_2$ is determinant of cofactor matrix of ${\Delta }_1$

$⟹{\Delta }_2={\Delta }_1^2$

Similarly ${\Delta }_3={\Delta }_2^2={\Delta }_1^4$

${\Delta }_1,{\Delta }_2,{\Delta }_3$ are in$G.P$ with common ratio ${\Delta }_1^2$

$⟹\frac{{\Delta }_3}{{\Delta }_2}={\Delta }_1^2$