Question

If $\begin{vmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}=5$ , then the value of $\begin{vmatrix} b_{2}c_{3}-b_{3}c_{2} & a_{3}c_{2}-a_{2}c_{3} & a_{2}b_{3}-a_{3}b_{2} \\ b_{3}c_{1}-b_{1}c_{3} & a_{1}c_{3}-a_{3}c_{1} & a_{3}b_{1}-a_{1}b_{3} \\ b_{1}c_{2}-b_{2}c_{1} & a_{2}c_{1}-a_{1}c_{2} & a_{1}b_{2}-a_{2}b_{1} \end{vmatrix}$ is

• A. $5$
• B. $25$
• C. $125$
• D. $0$

Answer 1

Answer: (B)

Let $\Delta =\begin{vmatrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{vmatrix}=5$ and $\Delta ^{'}=\begin{vmatrix} b_{2}c_{3}-b_{3}c_{2} & a_{3}c_{2}-a_{2}c_{3} & a_{2}b_{3}-a_{3}b_{2} \\ b_{3}c_{1}-b_{1}c_{3} & a_{1}c_{3}-a_{3}c_{1} & a_{3}b_{1}-a_{1}b_{3} \\ b_{1}c_{2}-b_{2}c_{1} & a_{2}c_{1}-a_{1}c_{2} & a_{1}b_{2}-a_{2}b_{1} \end{vmatrix}$
here $\Delta ^{'}$ is co-factor determinant of $\Delta $
hence $\Delta ^{'}=\Delta ^{n - 1}\Rightarrow \, \Delta ^{3 - 1}= \, \Delta ^{2}$
$= \, \left(5\right)^{2}= \, 25$